Low Density Phases in a Uniformly Charged Liquidmuratov/ndrop3dok.pdf · Low Density Phases in a Uniformly Charged Liquid ... which we will also refer to as the “self-energy problem”,

Digital Object Identifier (DOI) 10.1007/s00220-016-2654-3Commun. Math. Phys. 345, 141–183 (2016) Communications in

MathematicalPhysics

Low Density Phases in a Uniformly Charged Liquid

Hans Knüpfer1, Cyrill B. Muratov2, Matteo Novaga3

1 Institut für Angewandte Mathematik, Universität Heidelberg, INF 294, 69120 Heidelberg, Germany2 Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA.E-mail: [email protected]

3 Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

Received: 21 April 2015 / Accepted: 29 March 2016Published online: 26 May 2016 – © Springer-Verlag Berlin Heidelberg 2016

Abstract: This paper is concerned with the macroscopic behavior of global energyminimizers in the three-dimensional sharp interface unscreened Ohta–Kawasaki modelof diblock copolymer melts. This model is also referred to as the nuclear liquid dropmodel in the studies of the structure of highly compressed nuclear matter found inthe crust of neutron stars, and, more broadly, is a paradigm for energy-driven patternforming systems inwhich spatial order arises as a result of the competition of short-rangeattractive and long-range repulsive forces. Herewe investigate the large volume behaviorof minimizers in the low volume fraction regime, in which one expects the formation ofa periodic lattice of small droplets of the minority phase in a sea of the majority phase.Under periodic boundary conditions, we prove that the considered energy �-convergesto an energy functional of the limit “homogenized”measure associated with theminorityphase consisting of a local linear term and a non-local quadratic term mediated by theCoulomb kernel. As a consequence, asymptotically the mass of the minority phase in aminimizer spreads uniformly across the domain. Similarly, the energy spreads uniformlyacross the domain aswell, with the limit energy densityminimizing the energy of a singledroplet per unit volume. Finally, we prove that in the macroscopic limit the connectedcomponents of the minimizers have volumes and diameters that are bounded above andbelow by universal constants, and that most of them converge to the minimizers of theenergy divided by volume for the whole space problem.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1422. Mathematical Setting and Scaling . . . . . . . . . . . . . . . . . . . . . . 1463. Statement of the Main Results . . . . . . . . . . . . . . . . . . . . . . . . 1504. The Problem in the Whole Space . . . . . . . . . . . . . . . . . . . . . . 153

4.1 The truncated energy ˜ER∞ . . . . . . . . . . . . . . . . . . . . . . . . 1534.2 Generalized minimizers of ˜E∞ . . . . . . . . . . . . . . . . . . . . . 155

http://crossmark.crossref.org/dialog/?doi=10.1007/s00220-016-2654-3&domain=pdf

142 H. Knüpfer, C. B. Muratov, M. Novaga

4.3 Properties of the function e(m) . . . . . . . . . . . . . . . . . . . . . 1584.4 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5. Proof of Theorems 3.3 and 3.5 . . . . . . . . . . . . . . . . . . . . . . . . 1625.1 Compactness and lower bound . . . . . . . . . . . . . . . . . . . . . 1625.2 Upper bound construction . . . . . . . . . . . . . . . . . . . . . . . . 1655.3 Equidistribution of energy . . . . . . . . . . . . . . . . . . . . . . . 168

6. Uniform Estimates for Minimizers of the Rescaled Energy . . . . . . . . . 1697. Proof of Theorem 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177A. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

1. Introduction

The liquid dropmodel of the atomic nucleus, introduced byGamow in 1928, is a classicalexample of a model that gives rise to a geometric variational problem characterized bya competition of short-range attractive and long-range repulsive forces [5,6,30,68] (formore recent studies, see e.g. [17,18,51,52,57]; for a recent non-technical overviewof nuclear models, see, e.g., [19]). In a nucleus, different nucleons attract each othervia the short-range nuclear force, which, however, is counteracted by the long-rangeCoulomb repulsion of the constitutive protons. Within the liquid drop model, the effectof the short-range attractive forces is captured by postulating that the nucleons form anincompressible fluid with fixed nuclear density and by penalizing the interface betweenthe nuclear fluid and vacuum via an effective surface tension. The effect of Coulombrepulsion is captured by treating the nuclear charge as uniformly spread throughout thenucleus. A competition of the cohesive forces that try to minimize the interfacial area ofthe nucleus and the repulsive Coulomb forces that try to spread the charges apart makesthe nucleus unstable at sufficiently large atomic numbers, resulting in nuclear fission[6,22,27,47].

It is worth noting that the liquid drop model is also applicable to systems of manystrongly interacting nuclei. Such a situation arises in the case of matter at very highdensities, occurring, for example, in the core of a white dwarf star or in the crust of aneutron star, where large numbers of nucleons are confined to relatively small regions ofspace by gravitational forces [4,41,58]. As was pointed out independently by Kirzhnits,Abrikosov and Salpeter, at sufficiently low temperatures and not too high densitiescompressed matter should exhibit crystallization of nuclei into a body-centered cubiccrystal in a sea of delocalized degenerate electrons [1,38,63]. At yet higher densities,more exotic nuclear “pasta phases” are expected to appear as a consequence of the effectof “neutron drip” [35,43,44,55,56,58,59,65] (for an illustration, see Fig. 1). In all cases,the ground state of nuclear matter is determined by minimizing the appropriate (free)energy per unit volume of one of the phases that contains contributions from the interfacearea and the Coulomb energy of the nuclei.

Within the liquid drop model, the simplest way to introduce confinement is to restrictthe nuclear fluid to a bounded domain and impose a particular choice of boundaryconditions for the Coulombic potential. Then, after a suitable non-dimensionalizationthe energy takes the form

E(u) :=∫

�

|∇u| dx +1

2

∫

�

∫

�

G(x, y)(u(x) − u)(u(y) − u) dx dy. (1.1)

Here, � ⊂ R3 is the spatial domain (bounded), u ∈ BV (�; {0, 1}) is the characteristic

function of the region occupied by the nuclear fluid (nuclear fluid density), u ∈ (0, 1) is

Low Density Phases in a Uniformly Charged Liquid 143

Fig. 1. Nuclear pasta phases in a relativistic mean-field model of low density nuclear matter. The panels showa progression from “meatball” (a) to “spaghetti” (b) to “lasagna” (c) to “macaroni” (d) to “swiss cheese”(e) phases, which are the numerically obtained candidates for the ground state at different nuclear densities.Reproduced from Ref. [55], with permission

the neutralizing uniform background density of electrons, and G is the Green’s functionof the Laplacian, which in the case of Neumann boundary conditions for the electrostaticpotential solves

− �xG(x, y) = δ(x − y) − 1

|�| , (1.2)

where δ(x) is the Dirac delta-function. The nuclear fluid density must also satisfy theglobal electroneutrality constraint:

1

|�|∫

�

u dx = u. (1.3)

In writing (1.1) we took into account that because of the scaling properties of the Green’sfunction one can eliminate all the physical constants appearing in (1.1) by choosing theappropriate energy and length scales.

It is notable that the model in (1.1)–(1.3) also appears in a completely differentphysical context, namely, in the studies of mesoscopic phases of diblock copolymermelts, where it is referred to as the Ohta–Kawasaki model [14,54,60]. This is, of course,not surprising, considering the fundamental nature of Coulomb forces. In fact, the rangeof applications of the energy in (1.1) goes far beyond the systems mentioned above(for an overview, see [48] and references therein). Importantly, the model in (1.1) is a


paradigm for the energy-driven pattern forming systems in which spatial patterns (globalor local energy minimizers) form as a result of the competition of short-range attractiveand long-range repulsive forces. This is why this model and its generalizations attractedconsiderable attention of mathematicians in recent years (see, e.g., [2,3,7,10–13,16,23,26,32,33,36,37,39,40,45,49,50]; this list is certainly not exhaustive). In particular, thevolume-constrained global minimization problem for (1.1) in the whole space with noneutralizing background, which we will also refer to as the “self-energy problem”, hasbeen investigated in [12,26,40,45]. An asymptotic regime in which the minimizers ofthe energy in (1.1) concentrate into point masses in two and three space dimensions wasinvestigated in [12].

A question of particular physical interest is how the ground states of the energy in(1.1) behave as the domain size tends to infinity. In [3], it was shown that in this so-called“macroscopic” limit the energy becomes distributed uniformly throughout the domain.Another asymptotic regime, corresponding to the onset of non-trivial minimizers in thetwo-dimensional screened version of (1.1) was studied in [49], where it was shown thatat appropriately low densities every non-trivial minimizer is given by the characteristicfunction of a collection of nearly perfect, identical, well separated small disks (droplets)uniformly distributed throughout the domain (see also [32] for a related study of almostminimizers). Further results about the fine properties of theminimizerswere obtained viatwo-scale �-expansion in [33], using the approach developed for the studies of magneticGinzburg–Landau vortices [64] (more recently, the latter was also applied to three-dimensional Coulomb gases [62]). In particular, the method of [33] allows, in principle,to determine the asymptotic spatial arrangement of the droplets of the low density phasevia the solution of a minimization problem involving point charges in the plane. It iswidely believed that the solution of this problem should be given by a hexagonal lattice,which in the context of type-II superconductors is called the “Abrikosov lattice” [67].Proving this result rigorously is a formidable task, and to date such a result has beenobtained only within a much reduced class of Bravais lattices [9,64].

It is natural to ask what happens with the low density ground state of the energy in(1.1) as the size of the domain � goes to infinity in three space dimensions. As can beseen from the above discussion, the answer to this question bears immediate relevanceto the structure of nuclear matter under the conditions realized in the outer crust ofneutron stars. This is the question that we address in the present paper. On physicalgrounds, it is expected that at low densities the ground state of such systems is givenby the characteristic function of a union of nearly perfect small balls (nuclei) arrangedinto a body-centered cubic lattice (known to minimize the Coulomb energy of pointcharges among body-centered cubic, face-centered cubic and hexagonal close-packedlattices [24,28,53]). The volume of each nucleus should maximize the binding energyper nucleon, which then yields the nucleus of an isotope of nickel.

Our results concerning the minimizers of (1.1) proceed in that direction, but are stillfar from rigorously establishing such a detailed physical picture. Onemajor difficulty hasto dowith the lack of the complete solution of the self-energy problem [11,40].Assumingthe solution of this problem, whenever it exists, is a spherical droplet, a mathematicalconjecture formulated in [13] and a universally accepted hypothesis in nuclear physics,we indeed recover spherical nuclei whose volume minimizes the self-energy per unitnuclear volume (which is equivalent to maximizing the binding energy per nucleon inthe nuclear context). The question of spatial arrangement of the nuclei is another majordifficulty related to establishing periodicity of ground states of systems of interactingparticles, which goes far beyond the scope of the present paper. Nevertheless, knowing


that the optimal droplets are spherical should make it possible to apply the techniquesof [62,64] to relate the spatial arrangement of droplets to that of the minimizers of therenormalized Coulomb energy.

In the absence of the complete solution of the self-energy problem, we can still estab-lish, although in a somewhat implicit manner, the limit behavior of the minimizers of(1.1)–(1.3) in the case � = T�, where T� is the three-dimensional torus with sidelength�, as � → ∞, provided that u also goes simultaneously to zero with an appropriaterate (low-density regime). We do so by establishing the �-limit of the energy in (1.1),with the notion of convergence given by weak convergence of measures (for a closelyrelated study, see [32]). The limit energy is given by the sum of a constant term pro-portional to the volume occupied by the minority phase (which is also referred to as“mass” throughout the paper) and the Coulombic energy of the limit measure, with theproportionality constant in the first term given by the minimal self-energy per unit massamong all masses for which the minimum of the self-energy is attained. Importantly, theminimizer of the limit energy (which is strictly convex) is given uniquely by the uniformmeasure. Thus, we establish that for a minimizer of (1.1)–(1.3) the mass in the minorityphase spreads (in a coarse-grained sense) uniformly throughout the spatial domain andthat the minimal energy is proportional to the mass, with the proportionality constantgiven by the minimal self-energy per unit mass (compare to [3]). We also establish thatalmost all the “droplets”, i.e., the connected components of the support of a particularminimizer, are close to the minimizers of the self-energy with mass that minimizes theself-energy per unit mass.

Mathematically, it would be natural to try to extend our results in two directions. Thefirst direction is to consider exactly the same energy as in (1.1) in higher space dimen-sions. Here, however, we encounter a difficulty that it is not known that the minimizersof the self-energy do not exist for large enough masses. Such a result is only availablein three space dimensions for the Coulombic kernel [40,45]. In the absence of such anon-existence result one may not exclude a possibility of a network-like structure in themacroscopic limit. Another direction is to replace the Coulombic kernel in (1.1) withthe one corresponding to a more general negative Sobolev norm. Here we would expectour results to still hold in two space dimensions. Furthermore, the physical picture ofidentical radial droplets in the limit is expected for sufficiently long-ranged kernels, i.e.,those kernels that satisfyG(x, y) ∼ |x− y|−α for |x− y| � 1, with 0 < α � 1 [39,50].Note that although a similar characterization of the minimizers for long-ranged kernelsexists in higher dimensions as well [7,23], these results are still not sufficient to be usedto characterize the limit droplets, since they do not give an explicit interval of existenceof the minimizers of the self-interaction problem. Also, since the non-existence resultfor the self-energy with such kernels is available only for α < 2 [40], our results maynot extend to the case of α ≥ 2 in dimensions three and above.

Finally, a question of both physical and mathematical interest is what happens withthe above picture when the Coulomb potential is screened (e.g., by the backgrounddensity fluctuations). In the simplest case, one would replace (1.2) with the followingequation defining G:

− �xG(x, y) + κ2G(x, y) = δ(x − y), (1.4)

where κ > 0 is the inverse screening length, and the charge neutrality constraint from(1.3) is relaxed. Here a bifurcation from trivial to non-trivial ground states is expectedunder suitable conditions (in two dimensions, see [32,33,49]). We speculate that incertain limits this case may give rise to non-spherical droplets that minimize the self-


energy. Indeed, in the presence of an exponential cutoff at large distances, it may nolonger be advantageous to split large droplets into smaller disconnected pieces, and theself-energy minimizers for arbitrarily large masses may exist and resemble a “kebabon a skewer”. In contrast to the bare Coulomb case, in the screened case the energy ofsuch a kebab-shaped configuration scales linearly with mass. Note that this configura-tion is reminiscent of the pearl-necklace morphology exhibited by long polyelectrolytemolecules in poor solvents [20,25].

Organization of the paper In Sect. 2, we introduce the specificmodel, the scaling regimeconsidered, the functional setting and the heuristics. In this section, we also discuss theself-energy problem and mention a result about attainment of the optimal self-energyper unit mass. In Sect. 3, we first state a basic existence and regularity result for the min-imizers (Theorem 3.1) and give a characterization of the minimizers of the whole spaceproblem that also minimize the self-energy per unit mass (Theorem 3.2). We then stateourmain�-convergence result in Theorem3.3. In the same section, we also state the con-sequences of Theorem 3.3 to the asymptotic behavior of the minimizers in Corollary 3.4,as well as Theorem 3.5 about the uniform distribution of energy in the minimizers andTheorem 3.6 that establishes the multidroplet character of the minimizers. Section 4 isdevoted to generalized minimizers of the self-energy problem, where, in particular, weobtain existence and uniform regularity for minimizers in Theorems 4.5 and 4.7. Thissection also establishes a connection to the minimizers of the whole space problem witha truncated Coulombic kernel and ends with a characterization of the optimal self-energyper unit mass in Theorem 4.15. Section 5 contains the proof of the �-convergence resultof Theorem 3.3 and of the equidistribution result of Theorem 3.5. Section 6 establishesuniform estimates for the problem on the rescaled torus, where, in particular, uniformestimates for the potential are obtained in Theorem 6.9. Section 7 presents the proof ofTheorem 3.6. Finally, some technical results concerning the limit measures appearingin the �-limit are collected in the Appendix.

NotationThroughout the paper H1, BV , L p,Ck ,Ckc ,C

k,α ,M denote the usual spaces ofSobolev functions, functions of bounded variation, Lebesgue functions, functions withcontinuous derivatives up to order k, compactly supported functions with continuousderivatives up to order k, functions with Hölder-continuous derivatives up to order k forα ∈ (0, 1), and the space of finite signed Radon measures, respectively. We will use thesymbol |∇u| to denote the Radonmeasure associated with the distributional gradient of afunctionof boundedvariation.With a slight abuse of notation,wewill also identifyRadonmeasures with the associated, possibly singular, densities (with respect to the Lebesguemeasure) on the underlying spatial domain. For example, we will write ν = |∇u| anddν(x) = |∇u(x)| dx to imply ν ∈ M(�) and ν(�′) = |∇u|(�′) = ∫

�′ |∇u| dx , givenu ∈ BV (�) and �′ ⊂ �. For a set I ⊂ N, #I denotes the cardinality of I . The sym-bol χF always stands for the characteristic function of the set F , and |F | denotes itsLebesgue measure. We also use the notation (uε) ∈ Aε to denote sequences of functionsuε ∈ Aε as ε = εn → 0, where Aε are admissible classes.

2. Mathematical Setting and Scaling

Variational problem on the unit torusThroughout the rest of this paper the spatial domain� in (1.1) is assumed to be a torus,which allows us to avoid dealingwith boundary effectsand concentrate on the bulk properties of the energy minimizers. We define T := R

3/Z3

to be the flat three-dimensional torus with unit sidelength. For ε > 0, which should be


treated as a small parameter, we introduce the following energy functional:

Eε(u) := ε

∫

T

|∇u| dx +1

2

∫

T

(u − uε)(−�)−1(u − uε) dx, (2.1)

where the first term is understood distributionally and the second term is understoodas the double integral involving the periodic Green’s function of the Laplacian, with ubelonging to the admissible class

Aε :={

u ∈ BV (T; {0, 1}) :∫

T

u dx = uε

}

, (2.2)

where

uε := λ ε2/3, (2.3)

with some fixed λ > 0. The choice of the scaling of uε with ε in (2.3) will be explainedshortly. To simplify the notation, we suppress the explicit dependence of the admissibleclass on λ, which is fixed throughout the paper.

It is natural to define for u ∈ Aε the measure με by

dμε(x) := ε−2/3u(x) dx . (2.4)

In particular, με is a positive Radon measure and satisfies με(T) = λ. Therefore, on asuitable sequence as ε → 0 the measure με converges weakly in the sense of measuresto a limit measure μ, which is again a positive Radon measure and satisfies μ(T) = λ.

Function spaces for the measure and potential In terms of με the Coulombic term in(2.1) is given by

1

2

∫

T

(u − uε)(−�)−1(u − uε) dx = ε4/3

2

∫

T

∫

T

G(x − y) dμε(x) dμε(y), (2.5)

where G is the periodic Green’s function of the Laplacian on T, i.e., the unique distri-butional solution of

− �G(x) = δ(x) − 1,∫

T

G(x) dx = 0. (2.6)

If the kernel G in (2.5) were smooth, then one would be able to pass directly to the limitin the Coulombic term and obtain the corresponding convolution of the kernel with thelimit measure. This is not possible due to the singularity of the kernel at {x = y}. Infact, the double integral involving the limit measure may be strictly less than the lim infof the sequence, and the defect of the limit is related to a non-trivial contribution of theself-interaction of the connected components of the set {u = 1} and its perimeter to thelimit energy.

On the other hand, the singular character of the kernel provides control on the regular-ity of the limitmeasureμ. To see this, we define the electrostatic potential vε ∈ H1(T) by

vε(x) :=∫

T

G(x − y) dμε(y), (2.7)

which solves∫

T

∇ϕ · ∇vε dx =∫

T

ϕ dμε − λ

∫

T

ϕ dx ∀ϕ ∈ C∞(T). (2.8)


By (2.4), we can rewrite the corresponding term in the Coulombic energy as∫

T

∫

T

G(x − y) dμε(x) dμε(y) =∫

T

vε dμε =∫

T

|∇vε|2 dx . (2.9)

Hence, if the left-hand side of (2.9) remains bounded as ε → 0, and since∫

Tvε dx = 0,

the sequence vε is uniformly bounded in H1(T) and hence weakly convergent in H1(T)

on a subsequence.By the above discussion, the natural space for the potential is the space

H :={

v ∈ H1(T) :∫

T

v dx = 0

}

with ‖v‖H :=(∫

T

|∇v|2 dx)1/2

. (2.10)

The space H is a Hilbert space together with the inner product

〈u, v〉H :=∫

T

∇u · ∇v dx ∀u, v ∈ H. (2.11)

The natural class for measures με to consider is the class of positive Radon measures onT which are also in H′, the dual of H. More precisely, let M+(T) ⊂ M(T) be the setof all positive Radon measures on T. We define the subset M+(T) ∩ H′ of M+(T) by

M+(T) ∩ H′ ={

μ ∈ M+(T) :∫

T

ϕ dμ ≤ C‖ϕ‖H ∀ϕ ∈ H ∩ C(T)}

. (2.12)

This is the set of positive Radon measures which can be understood as continuous linearfunctionals onH. Note that μ ∈ M+(T) satisfies μ ∈ M+(T) ∩H′ if and only if it hasfinite Coulombic energy, i.e.

∫

T

∫

T

G(x − y) dμ(x) dμ(y) < ∞, (2.13)

with the convention that G(0) = +∞. The proof of this characterization and relatedfacts about M+(T) ∩ H′ are given in the Appendix.

The whole space problem We will also consider the following related problem, formu-lated on R

3. We consider the energy

˜E∞(u) :=∫

R3|∇u| dx +

1

8π

∫

R3

∫

R3

u(x)u(y)

|x − y| dx dy. (2.14)

The appropriate admissible class for the energy ˜E∞ in the present context is that ofconfigurations with prescribed “mass” m > 0:

Ã∞(m) :={

u ∈ BV (R3; {0, 1}) :∫

R3u dx = m

}

. (2.15)

For a given mass m > 0, we define the minimal energy by

e(m) := infu∈Ã∞(m)

˜E∞(u). (2.16)

The set of masses for which the infimum of ˜E∞ in Ã∞(m) is attained is denoted by

I := {

m ≥ 0 : ∃ um ∈ Ã∞(m), ˜E∞(um) = e(m)}

, (2.17)


Theminimization problem associated with (2.14) and (2.15) was recently studied by twoof the authors in [40]. In particular, by [40, Theorem 3.3] the set I is bounded, and by [40,Theorems 3.1 and 3.2] the set I is non-empty and contains an interval around the origin.

For m ≥ 0, we also define the quantity (with the convention that f (0) := +∞)

f (m) := e(m)

m, (2.18)

which represents the minimal energy for (2.14) and (2.15) per unit mass. By [40, The-orem 3.2] there is a universal m0 > 0 such that f (m) is obtained by evaluating ˜E∞ ona ball of mass m for all m ≤ m0. After a simple computation, this yields

f (m) = 62/3π1/3m−1/3 + 32/3 · 2−1/3 · 10−1 · π−2/3m2/3 for all 0 < m ≤ m0.

(2.19)

Note that obviously this expression also gives an a priori upper bound for f (m) for allm > 0. In addition, by [40, Theorem 3.4] there exist universal constants C, c > 0 suchthat

c ≤ f (m) ≤ C for all m ≥ m0. (2.20)

It was conjectured in [13] that I = [0, m0] and that m0 = mc1, where

mc1 := 40π

3

(

21/3 + 2−1/3 − 1)

≈ 44.134. (2.21)

The quantity mc1 is the maximum value of m for which a ball of mass m has less energythan twice the energy of a ball with mass 1

2m. However, such a result is not availableat present and remains an important challenge for the considered class of variationalproblems (for several related results see [7,39,50]).

Finally, we define

f ∗ := infm∈I

f (m) and I∗ := {

m∗ ∈ I : f (m∗) = f ∗} . (2.22)

Observe that in view of (2.19) and (2.20) we have f ∗ ∈ (0,∞). Also, as we will showin Theorem 3.2, the set I∗ is non-empty, i.e., the minimum of f (m) over I is attained.In fact, the minimum of f (m) over I is also the minimum over all m ∈ (0,∞) (seeTheorem 4.15). Note that this result was also independently obtained by Frank and Liebin their recent work [26]. The set I∗ of masses that minimize the energy ˜E∞ per unitmass and the associated minimizers (which in general may not be unique) will play akey role in the analysis of the limit behavior of the minimizers of Eε. Note that if f (m)

were given by (2.19) and I = [0,mc1], then we would have explicitly I∗ = {10π} andf ∗ = 35/3·2−2/3·5−1/3 ≈ 2.29893.On the other hand, in viewof the statement following(2.19), this value provides an a priori upper bound on the optimal energy density.

Macroscopic limit and heuristics The limit ε → 0 with λ > 0 fixed is equivalent tothe limit of the energy in (1.1) with � = T�, where T� := R

3/(�Z)3 is the torus withsidelength � > 0, as � → ∞. Indeed, introducing the notation

˜E�(u) :=∫

T�

|∇u| dx +1

2

∫

T�

(u − ¯u�)(−�)−1(u − ¯u�) dx, (2.23)


for the energy in (1.1) with � = T�, and taking ¯u� = λ�−2 and u ∈ Ã�, where

Ã� :={

u ∈ BV (T�; {0, 1}) :∫

T�

u dx = λ�

}

, (2.24)

it is easy to see that u(x) := u(�x) belongs to Aε with uε = λε2/3 for ε = �−3, andwe have Eε(u) = �−5

˜E�(u). It then follows that the two limits � → ∞ and ε → 0 areequivalent. Note that the full space energy ˜E∞ is the formal limit of (2.23) for � → ∞.

The choice of the scaling of uε with ε is determined by the balance of far-field andnear-field contributions of the Coulomb energy. Heuristically, one would expect theminimizers of the energy in (2.1) to be given by the characteristic function of a set thatconsists of “droplets” of size of order R � 1 separated by distance of order d satisfyingR � d � 1 (for evidence based on recent molecular dynamics simulations, see also[65]). Assuming that the volume of each droplet scales as R3 (think, for example, of allthe droplets as non-overlapping balls of equal radius and with centers forming a periodiclattice), from (2.23) we find for the surface energy, self-energy and interaction energy,respectively, for a single droplet:

Esurf ∼ εR2, Eself ∼ R5, Eint ∼ R6

d3. (2.25)

Equating these three quantities and recalling that R3/d3 ∼ uε, we obtain

R ∼ ε1/3, d ∼ ε1/9, uε ∼ ε2/3, (2.26)

which leads to (2.3). Note that, in some sense, this is the most interesting low volumefraction regime that leads to infinitely many droplets in the limit as ε → 0, since both theself-energy of each droplet and the interaction energy between different droplets con-tribute comparably to the energy. For other scalings one would expect only one of thesetwo terms to contribute in the limit, which would, however, result in loss of control oneither the perimeter termor theCoulomb term as ε → 0 and, as a consequence, a possiblechange in behavior. Let us note that a different scaling regime, in which uε = O(ε2/3)

and � = O(ε1/9), leads instead to finitely many droplets that concentrate on points asε → 0 [12], while for uε = O(1) one expects phases of reduced dimensionality, suchas rods and slabs (see Fig. 1).

3. Statement of the Main Results

We now turn to stating the main results of this paper concerning the asymptotic behaviorof theminimizers or the low energy configuration of the energy in (2.1) within the admis-sible class in (2.2). Existence of theseminimizers is guaranteed by the following theorem.

Theorem 3.1 (Minimizers: existence and regularity). For every λ > 0 and every0 < ε < λ−3/2 there exists a minimizer uε ∈ Aε of Eε given by (2.1) with uε givenby (2.3). Furthermore, after a possible modification of uε on a set of zero Lebesguemeasure the support of uε has boundary of class C∞.

Proof. The proof of Theorem 3.1 is fairly standard. We present a few details below forthe sake of completeness.

By the direct method of the calculus of variations, minimizers of the consideredproblem exist for all ε > 0 as soon as the admissible class Aε is non-empty, in view


of the fact that the first term is coercive and lower semicontinuous in BV (T), and thatthe second term is continuous with respect to the L1(T) convergence of characteristicfunctions. The admissible class is non-empty if and only if ε < λ−3/2.

Hölder regularity of minimizers was proved in [66, Proposition 2.1], where it wasshown that the essential support of minimizers has boundary of class C3,α . Smoothnessof the boundarywas established in [37, Proposition 2.2] (see also the proof of Lemma 4.4below for a brief outline of the argument in a closely related context). ��

In view of the regularity statement above, throughout the rest of the paper we alwayschoose the regular representative of a minimizer.

We proceed by giving a characterization of the quantity f ∗ defined in (2.22) as theminimal self-energy of a single droplet per unit mass, i.e., as the minimum of f (m) overI.Theorem 3.2 (Self-energy: attainment of optimal energy per unit mass). Let f ∗ bedefined as in (2.22). Then there exists m∗ ∈ I such that f ∗ = f (m∗).

With the result in Theorem 3.2, we are now in the position to state our main result on the�-limit of the energy in (2.1),which can be viewed as a generalization of [32, Theorem1].

Theorem 3.3 (�-convergence). For a given λ > 0, let Eε be defined by (2.1) with uε

given by (2.3). Then as ε → 0 we have ε−4/3Eε�→ E0, where

E0(μ) := λ f ∗ + 1

2

∫

T

∫

T

G(x − y) dμ(x) dμ(y), (3.1)

and μ ∈ M+(T) ∩ H′ satisfies μ(T) = λ. More precisely,

(i) (Compactness and �-liminf inequality) Let (uε) ∈ Aε be such that

lim supε→0

ε−4/3Eε(uε) < ∞, (3.2)

and let με and vε be defined in (2.4) and (2.7), respectively. Then, upon extractionof a subsequence, we have

με ⇀ μ inM(T), vε ⇀ v inH, (3.3)

as ε → 0, for some μ ∈ M+(T) ∩ H′ with μ(T) = λ, the function v has a repre-sentative in L1(T, dμ) given by

v(x) =∫

T

G(x − y) dμ(y), (3.4)

and

lim infε→0

ε−4/3Eε(uε) ≥ E0(μ). (3.5)

(ii) (�-limsup inequality) For any measure μ ∈ M+(T)∩H′ with μ(T) = λ there existsa sequence (uε) ∈ Aε such that (3.3) and (3.4) hold as ε → 0 for με and vε definedin (2.4) and (2.7), and

lim supε→0

ε−4/3Eε(uε) ≤ E0(μ). (3.6)


Note that the weak convergence of measures was recently identified in [32] (seealso [49]) as a suitable notion of convergence for the studies of the �-limit of the two-dimensional version of the energy in (2.1).

Observe also that the limit energy E0 is a strictly convex functional of the limit mea-sure and, hence, attains a unique global minimum. By direct inspection, E0 is minimizedby μ = μ0, where dμ0 := λdx . Thus, the quantity f ∗ plays the role of the optimalenergy density in the limit ε → 0.

The remaining results are concerned with sequences of minimizers. We will henceassume that the functions (uε) ∈ Aε are minimizers of the functional Eε. In this case,we can give a more precise characterization for the asymptotic behavior of the sequence.We first note the following immediate consequence of Theorem 3.3 for the convergenceof sequences of minimizers.

Corollary 3.4 (Minimizers: uniform distribution of mass). For λ > 0, let (uε) ∈ Aε beminimizers of Eε, and let με and vε be defined in (2.4) and (2.7), respectively. Then

με ⇀ μ0 inM(T), vε ⇀ 0 inH, (3.7)

where dμ0 = λdx, and

ε−4/3Eε(uε) → λ f ∗, (3.8)

where f ∗ is as in (2.22), as ε → 0.

The formula in (3.8) suggests that in the limit the energy of the minimizers is domi-nated by the self-energy, which is captured by the minimization problem associated withthe energy ˜E∞ defined in (2.14). Therefore, it would be natural to expect that asymptoti-cally every connected component of a minimizer is close to a minimizer of ˜E∞ under themass constraint associated with that connected component. Note that in a closely relatedproblem in two space dimensions such a result was established in [49] for minimizers,and in [32,33] for almost minimizers. The situation is, however, unique in two spacedimensions, because the non-local term in some sense decouples from the perimeterterm. Hence, the minimizers behave as almost minimizers of the perimeter and, there-fore, are close to balls. In three dimensions, however, the perimeter and the non-localterm of the self-energy ˜E∞ are fully coupled, and, therefore, rigidity estimates for theperimeter functional alone [29] may not be sufficient to conclude about the “shape” ofthe minimizers. Nevertheless, we are able to prove a result about the uniform distribu-tion of the energy density of the minimizers as ε → 0 in the spirit of that of [3]. For aminimizer uε, the energy density is associated with the Radon measure νε defined by

dνε := ε−4/3(

ε|∇uε| + 12ε

2/3uεvε

)

dx, (3.9)

where vε is given by (2.7) and (2.4). Furthermore, we are able to identify the leadingorder constant in the asymptotic behavior of the energy density.

Theorem 3.5 (Minimizers: uniform distribution of energy). For λ > 0, let (uε) ∈ Aε

be minimizers of Eε and let νε be defined in (3.9). Then

νε ⇀ ν0 inM(T) as ε → 0, (3.10)

where dν0 = λ f ∗dx and f ∗ is as in (2.22).


Finally, we characterize the connected components of the support of the minimizersof Eε and show that almost all of them approach, on a suitable sequence as ε → 0 andafter a suitable rescaling and translation, a minimizer of ˜E∞ with mass in the set I∗.

Theorem 3.6 (Minimizers: droplet structure). For λ > 0, let (uε) ∈ Aε be regular rep-resentatives of minimizers of Eε, let Nε be the number of the connected components ofthe support of uε, let uε,k ∈ BV (R3; {0, 1}) be the characteristic function of the k-thconnected component of the support of the periodic extension of uε to the whole of R

3

modulo translations in Z3, and let xε,k ∈ supp(uε,k). Then there exists ε0 > 0 such that

the following properties hold:(i) There exist universal constants C, c > 0 such that for all ε ≤ ε0 we have

‖vε‖L∞(T) ≤ C and∫

R3uε,k dx ≥ cε, (3.11)

where vε is given by (2.7).(ii) There exist universal constants C, c > 0 such that for all ε ≤ ε0 we have

supp(uε,k) ⊆ BCε1/3(xε,k) and cλε−1/3 ≤ Nε ≤ Cλε−1/3. (3.12)

(iii) There exists Ñε ≤ Nε with Ñε/Nε → 1 as ε → 0 and a subsequence εn → 0 suchthat for every kn ≤ Ñεn the following holds: After possibly relabeling the connectedcomponents, we have

un → u in L1(R3), (3.13)

where un(x) := uεn ,kn (ε1/3n (x + xεn ,kn )), and u is a minimizer of ˜E∞ over Ã∞(m∗)

for some m∗ ∈ I∗, where I∗ is defined in (2.22).

The significance of this theorem lies in the fact that it shows that all the connectedcomponents of the support of aminimizer for sufficiently small ε look like a collection ofdroplets of size of order ε1/3 separated by distances of order ε1/9 on average. In particular,the conclusion of the theorem excludes configurations that span the entire length of thetorus, such as the “spaghetti” or “lasagna” phases of nuclear pasta (see Fig. 1). Thus, theground state for small enough ε > 0 is a multi-droplet pattern (a “meatball” phase). Fur-thermore, after a rescalingmost of these droplets converge tominimizers of the non-localisoperimetric problem associated with ˜E∞ that minimize the self-energy per unit mass.

4. The Problem in the Whole Space

In this section, we derive some results about the single droplet problem from (2.14)–(2.15).

4.1. The truncated energy ˜ER∞. For reasons that will become apparent shortly, it is help-ful to consider the energies where the range of the nonlocal interaction is truncated atcertain length scale R. We choose a cut-off function η ∈ C∞(R) with η′(t) ≤ 0 forall t ∈ R, η(t) = 1 for all t ≤ 1 and η(t) = 0 for all t ≥ 2. In the following, thechoice of η is fixed once and for all, and the dependence of constants on this choice is


suppressed to avoid clutter in the presentation. For R > 0, we then define ηR ∈ C∞(R3)

by ηR(x) := η(|x |/R). For u ∈ Ã∞(m), we consider the truncated energy

˜ER∞(u) :=∫

R3|∇u| dx +

∫

R3

∫

R3

ηR(x − y)u(x)u(y)

8π |x − y| dx dy. (4.1)

This functional will be useful in the analysis of the variational problems associated with˜E∞ and Eε. We recall that by the results of [61], for each R > 0 and each m > 0 thereexists a minimizer of ˜ER∞ in Ã∞(m). Furthermore, after a possible redefinition on a setof Lebesgue measure zero, its support has boundary of class C1,α for any α ∈ (0, 1

2 ),and consists of finitely many connected components. Below we always deal with therepresentatives of minimizers that are regular.

The following uniform density bound for minimizers of the energy is an adaptation of[40, Lemma 4.3] for the truncated energy ˜ER∞ and generalizes the corresponding boundfor minimizers of ˜E∞.

Lemma 4.1 (Density bound). There exists a universal constant c > 0 such that for everyminimizer u ∈ Ã∞(m) of ˜ER∞ for some R,m > 0 and any x0 ∈ F we have

∫

Br (x0)u dx ≥ cr3 for all r ≤ min(1,m1/3). (4.2)

Proof. The claim follows by an adaption of the proof of [40, Lemma 4.3] to our truncatedenergy ˜ER∞. Indeed, it is enough to show that the statement of [40, Lemma 4.2] holdswith ˜E∞ replaced by ˜ER∞. The proof of this statement needs to be modified, since thekernel in the definition of ˜ER∞ is not scale-invariant. We sketch the necessary changes,using the same notation as in [40].

The construction of the sets ˜F and F proceeds as in the proof of [40, Lemma 4.3].The upper bound [40, Eq. (4.6)] still holds since ˜ER∞(u) ≤ ˜E∞(u). Related to the cut-offfunction in the definition of ˜ER∞, we get an additional term in the right-hand side of thefirst line of [40, Eq. (4.6)], which is of the form

∫

�F1

∫

�F1

ηR(x − y)

|x − y|α dx dy − �2n−α

∫

F1

∫

F1

ηR(x − y)

|x − y|α dx dy

= �2n−α

∫

F1

∫

F1

ηR/�(x − y) − ηR(x − y)

|x − y|α dx dy < 0, (4.3)

since � > 1 and since the function η is monotonically decreasing (note that α = 1 inour case). Since this term has a negative sign, [40, Eq. (4.6)] still holds. The rest of theargument then carries through unchanged. ��

The following lemma establishes a uniform diameter bound for the minimizers of˜ER∞. The idea of the proof is similar to the one in [45, Lemma 5].

Lemma 4.2 (Diameter bound). There exist universal constants R0 > 0 and D0 > 0such that for any R ≥ R0, any m > 0 and for any minimizer u ∈ Ã∞(m) of˜ER∞, the diameter of each connected component F0 of supp(u) is bounded aboveby D0.

Proof. Let F0 be a connected component of the support of u withm0 := |F0|. Since u is aminimizer, χF0 is also a minimizer of ˜ER∞ overA∞(m0). Indeed, if not, replacing u with


u −χF0 +χ˜F0 , where χ

˜F0 is a minimizer of ˜ER∞ overA∞(m0) translated sufficiently farfrom the support of uwould lower the energy, contradicting theminimizing property of u.

Wemay assumewithout loss of generality that R ≥ 2 and diam F0 ≥ 2. Then there isN ∈ N such that 2N ≤ diam F0 < 2(N + 1). In particular there exist x0, . . . , xN ∈ F0such that |xk − x0| = 2k for every 1 ≤ k ≤ N and, therefore, the balls B1(xk) aremutually disjoint. If m0 ≤ 1, then by Lemma 4.1 we have |F0 ∩ Br (xk)| ≥ cm0 forr = m1/3

0 ≤ 1 and some universal c > 0. Therefore,

m0 ≥N

∑

k=1

|F0 ∩ Br (xk)| ≥ cm0N , (4.4)

implying that N ≤ N0 for some universal N0 ≥ 1 and, hence, diam F0 ≤ 2(N0 + 1).If, on the other hand, m0 > 1, then by Lemma 4.1 we have |F0 ∩ B1(xk)| ≥ c for someuniversal c > 0. By monotonicity of the kernel in |x − y|, we get∫

F0∩B1(x0)

∫

F0\B1(x0)ηR(x − y)

|x − y| dx dy ≥ c2N

∑

k=1

ηR(2k + 2)

2k + 2≥ C min{log N , log R},

for some universal C > 0. Hence, if R and N are sufficiently large, then it is energeti-cally preferable to move the charge in B1(x0) sufficiently far from the remaining charge.More precisely, consider u = u − χF0∩B1(x0) + χF0∩B1(x0)(· + b), for some b ∈ R

3 with|b| sufficiently large. Then u ∈ A∞(m0) and

˜ER∞(u) ≤ ˜ER∞(u) + 4π − 12C min{log N , log R} < 0, (4.5)

for all R ≥ R0 and N > N0 for some universal constants R0 ≥ 2 and N0 ≥ 1. Therefore,minimality of u implies that N ≤ N0 whenever R ≥ R0 and hence diam F0 ≤ 2(N0+1).

��

4.2. Generalized minimizers of ˜E∞. We begin our analysis of ˜E∞ by introducing thenotion of generalized minimizers of the non-local isoperimetric problem.

Definition 4.3 (Generalized minimizers). Givenm > 0, we call a generalized minimizerof ˜E∞ in Ã∞(m) a collection of functions (u1, . . . , uN ) for some N ∈ N such that uiis a minimizer of ˜E∞ over Ã∞(mi ) with mi = ∫

Tui dx for all i ∈ {1, . . . , N }, and

m =N

∑

i=1

mi and e(m) =N

∑

i=1

e(mi ). (4.6)

Clearly, everyminimizer of ˜E∞ in Ã∞(m) is also a generalizedminimizer (with N = 1).As was shown in [40], however, minimizers of ˜E∞ in Ã∞(m) may not exist for a givenm > 0 because of the possibility of splitting their support into several connected com-ponents and moving those components far apart. As we will show below, this possibleloss of compactness of minimizing sequences can be compensated by considering char-acteristic functions of sets whose connected components are “infinitely far apart” andamong which the minimum of the energy is attained (by a generalized minimizer withsome N > 1). We also remark that, if (u1, . . . , uN ) is a generalized minimizer, then, as


can be readily seen from the definition, any sub-collection of ui ’s is also a generalizedminimizer with the mass equal to the sum of the masses of its components.

We now proceed to demonstrating existence of generalized minimizers of ˜E∞ for allm > 0. We start by stating the basic regularity properties of the minimizers of ˜E∞ andthe associated Euler–Lagrange equation.

Lemma 4.4 (Regularity and Euler–Lagrange equation). For m > 0, let u be a minimizerof ˜E∞ in Ã∞(m), and let F = supp (u). Then, up to a set of Lebesgue measure zero,the set F is a bounded connected set with boundary of class C∞, and we have

2κ(x) + vF (x) = λF for x ∈ ∂F, (4.7)

where λF ∈ R is a Lagrange multiplier, κ(x) is the mean curvature of ∂F at x (positiveif F is convex), and

vF (x) := 1

4π

∫

F

dy

|x − y| . (4.8)

Moreover, if m ∈ [m0,m1] for some 0 < m0 < m1, then vF ∈ C1,α(R3) and ∂F is ofclass C3,α, for all α ∈ (0, 1), uniformly in m.

Proof. From [40, Proposition 2.1 and Lemma 4.1] it follows that, up to a set of Lebesguemeasure zero, the set F is bounded and connected, and ∂F is of class C1,α for anyα ∈ (0, 1

2 ). Since the function vF is the unique solution of the elliptic problem−�v = χF with v(x) → 0 for |x | → ∞, by [40, Lemma 4.4] and elliptic regularity the-ory [31] it follows that vF ∈ C1,α(R3) for allα ∈ (0, 1), uniformly inm ∈ [m0,m1]. TheEuler–Lagrange equation (4.7) can be obtained as in [15, Theorem2.3] (see also [48,66]).Further regularity of ∂F follows from [66, Proposition 2.1] and [37, Proposition 2.2].

��Similarly, if (u1, . . . , uN ) is a generalized minimizer of ˜E∞ and Fi := supp (ui ) for

i ∈ {1, . . . , N }, the following Euler–Lagrange equation holds:

2κi (x) +1

4π

∫

Fi

dy

|x − y| = λ x ∈ ∂Fi , (4.9)

where κi is the mean curvature of ∂Fi (positive if Fi is convex) and λ ∈ R is a Lagrangemultiplier independent of i . The latter follows from the fact that generalized mini-mizers are easily seen to minimize

∑Ni=1

˜E∞(ui ) over all ui ∈ Ã∞(mi ) subject to∑N

i=1 mi = m.In contrast to minimizers, generalized minimizers of ˜E∞ in Ã∞(m) exist for all

m > 0:

Theorem 4.5 (Existence of generalized minimizers). For any m ∈ (0,∞) there existsa generalized minimizer (u1, . . . , uN ) of ˜E∞ in Ã∞(m). Moreover, after a possiblemodification on a set of Lebesgue measure zero, the support of each component ui isbounded, connected and has boundary of class C∞.

Proof. Wemay assume thatm ≥ m0, where m0 > 0 was defined in Sect. 2, since other-wise the minimum of ˜E∞ is attained by a ball [40, Theorem 3.2] and the statement of thetheorem holds true. In [61, Theorems 5.1.1 and 5.1.5], it is proved that the functional ˜ER∞admits a minimizer u = χFR ∈ Ã∞(m), FR ⊂ R

3, for any R > 0, and after a possibleredefinition on a set of Lebesguemeasure zero, the set FR is regular, in the sense that it is a


union of finitelymany connected components whose boundaries are of classC1,α for anyα ∈ (0, 1

2 ). Let F1, . . . , FN ⊂ R3 be the connected components of FR . By Lemma 4.1,

we have N ≤ N0, |Fk | ≥ δ0 and diam Fk ≤ D0 for all 1 ≤ k ≤ N and for some N0 ≥ 1and some constants D0, δ0 > 0 depending only on m. Furthermore, we have

dist(Fi , Fj ) ≥ 2R for i �= j, (4.10)

since otherwise it would be energetically preferable to increase the distance between thecomponents. In particular, if R ≥ D0 the family of sets F1, . . . , FN ⊂ R

3 generates ageneralized minimizer (u1, . . . , uN ) of ˜E∞ by letting ui := χFi . Indeed, we have

e(m) ≥ inf|F |=m˜ER∞(u) =

N∑

i=1

˜ER∞(ui ) =N

∑

i=1

˜E∞(ui ) ≥N

∑

i=1

e(|Fi |) ≥ e(m), (4.11)

and so all the inequalities in (4.11) are in fact equalities. Since ˜E∞(χFi ) ≥ e(|Fi |) foreach 1 ≤ i ≤ N , from (4.11)weobtain that each set Fi is aminimizer of ˜E∞ in Ã∞(|Fi |).By Lemma 4.4, each set Fi is bounded and connected, and ∂Fi are of class C∞. ��

The arguments in the proof of the previous theorem in fact show the following relationbetween minimizers of the truncated energy ˜ER∞ and generalized minimizers of ˜E∞.

Corollary 4.6 (Generalized minimizers as minimizers of the truncated problem). Letm > 0 and R > 0, let u ∈ Ã∞(m) be a minimizer of ˜ER∞, and let u = ∑N

i=1 ui , whereui are the characteristic functions of the connected components of the support of u. Thenthere exists a universal constant R1 > 0 such that if R ≥ R1, then (u1, . . . , uN ) is ageneralized minimizer of ˜E∞ in Ã∞(m).

Proof. We choose R1 = max{R0, D0}, where R0 and D0 are as in Lemma 4.2. Then wehave ˜ER∞(χF0) = ˜E∞(χF0) for every connected component F0 of the minimizer. Withthe same argument as the one used in the proof of Theorem 4.5, this yields the claim.

��We now provide some uniform estimates for generalized minimizers.

Theorem 4.7 (Uniform estimates for generalized minimizers). There exist universalconstants δ0 > 0 and D0 > 0 such that, for any m > m0, where m0 is defined in Sect. 2,the support of each component of a generalized minimizer of ˜E∞ in Ã∞(m) has volumebounded below by δ0 and diameter bounded above by D0 (after possibly modifying thecomponents on sets of Lebesgue measure zero).Moreover, there are universal constantsC, c > 0 such that the number N of the components satisfies

cm ≤ N ≤ Cm. (4.12)

Proof. Letm ≥ m0 and let (χF1 , . . . , χFN )be a generalizedminimizer of ˜E∞ in Ã∞(m),taking all sets Fi to be regular. By [40, Theorem 3.3] we know that there exists a universalm2 ≥ m0 such that

|Fi | ≤ m2 for all i ∈ {1, . . . , N }. (4.13)

Then by [40, Lemma 4.3] and the argument of [40, Lemma 4.1] we have

diam(Fi ) ≤ D0, (4.14)


for some universal D0 > 0. On the other hand, we claim that taking R ≥ D0 we have that

u(x) :=N

∑

i=1

χFi (x + 4i Re1), (4.15)

where e1 is the unit vector in the first coordinate direction, is a minimizer of ˜ER∞ inÃ∞(m). Indeed, since the connected components of the support of u are separated bydistance 2R, we have

˜ER∞(u) =N

∑

i=1

˜ER∞(χFi ) =N

∑

i=1

˜E∞(χFi ) = e(m). (4.16)

At the same time, by the argument in theproof ofTheorem4.5wehave infu∈Ã∞(m)˜ER∞(u) =

e(m) for all R sufficiently large depending on m. Hence, u is a minimizer of ˜ER∞ inÃ∞(m) for large enough R. The universal lower bound |Fi | ≥ δ0 then follows fromLemma 4.1 and our assumption on m.

Finally, the lower bound in (4.12) is a consequence of (4.13), while the upper boundfollows directly from the lower bound on the volume of the components just obtained.

��

4.3. Properties of the function e(m). In this section, we discuss the properties of thefunctions e(m) = infu∈ Ã∞(m)

˜E∞(u) and f (m) = e(m)/m, in particular their depen-dence on m.

We start by showing that e(m) is locally Lipschitz continuous on (0,∞).

Lemma 4.8 (Lipschitz continuity of e). The function e(m) is Lipschitz continuous oncompact subsets of (0,∞).

Proof. Let m,m′ ∈ [m0,m1] ⊂ (0,∞) and let (u1, . . . , uN ) be a generalized min-imizer of ˜E∞ in Ã∞(m). For λ = (m′/m)1/3, we define the rescaled functions uλ

iwith uλ

i (x) = ui (λ−1x). For sufficiently large R > 0, we define uλ ∈ Ã∞(m′) by

uλ(x) := ∑Ni=1 ui (λ

−1x + i Re1), where e1 is the unit vector in the first coordinatedirection. We then have

˜E∞(uλ) = λ2N

∑

i=1

∫

R3|∇ui | dx + λ5

N∑

i=1

∫

R3

∫

R3

ui (x)ui (y)

8π |x − y| dx dy + g(R), (4.17)

where the term g(R) refers to the interaction energy between different components uλi ,

uλj , i �= j , of uλ. Clearly, we have g(R) → 0 for R → ∞. It follows that

˜E∞(uλ) − e(m) ≤∣

∣

∣λ2 − 1

∣

∣

∣

N∑

i=1

∫

R3|∇ui | dx +

∣

∣

∣λ5 − 1

∣

∣

∣

N∑

i=1

∫

R3

∫

R3

ui (x)ui (y)

8π |x − y| dx dy + g(R). (4.18)

In the limit R → ∞, this yields e(m′) ≤ ˜E∞(uλ) ≤ e(m)(1+C |m−m′|) for a constantC > 0 that depends only onm0, m1. Sincem,m′ are arbitrary and since e(m) is bounded


above by the energy of a ball of mass m1, it follows that e is Lipschitz continuous on[m0,m1] for all 0 < m0 < m1. ��

We next establish a compactness result for generalized minimizers.

Lemma 4.9 (Compactness for generalized minimizers). Let mk be a sequence of posi-tive numbers converging to some m > m0, where m0 is defined in Sect. 2, as k → ∞,

and let (uk,1, . . . , uk,Nk ) be a sequence of generalized minimizers of ˜E∞ in Ã∞(mk).Then, up to extracting a subsequence we have that Nk = N ∈ N for all k, and aftersuitable translations uk,i ⇀ ui in BV (R3) as k → ∞ for all i ∈ {1, . . . , N }, where(u1, . . . , uN ) ∈ Ã∞(m) is a generalized minimizer of ˜E∞ in Ã∞(m).

Proof. By Theorem 4.7, we know that Nk ≤ M ∈ N for all k large enough. Hence, uponextraction of a subsequence we can assume that Nk = N for all k, for some N ∈ N. Forany i ∈ {1, . . . , N }, we also have

supk

∫

R3|∇uk,i | dx ≤ sup

m∈I˜E∞(χBm1/3 ) < ∞. (4.19)

Moreover, again by Theorem 4.7 we have mk,i ≥ δ0 and supp(uk,i ) ⊂ BD0(0), aftersuitable translations. Hence, up to extracting a further subsequence, there exist mi ≥ δ0and ui ∈ Ã∞(mi ) such that mk,i → mi and uk,i ⇀ ui in BV (R3), as k → ∞. Passingto the limit in the equalitiesmk = ∑N

i=1 mk,i and e(mk) = ∑Ni=1 e(mk,i ), we obtain that

m =N

∑

i=1

mi and e(m) =N

∑

i=1

e(mi ), (4.20)

where we used Lemma 4.8 to establish the last equality. Finally, again by Lemma 4.8and by lower semicontinuity of ˜E∞ we have e(mi ) ≤ ˜E∞(ui ) ≤ lim infk→∞ e(mk,i ) =e(mi ), which yields the conclusion. ��

With the two lemmas above, we are now in a position to prove the main result of thissubsection.

Lemma 4.10. The set I defined in (2.17) is compact.

Proof. Since I is bounded by [40, Theorem 3.3], it is enough to prove that it is closed.Let mk → m > 0, with mk ∈ I, and let uk ∈ Ã∞(mk) be such that ˜E∞(uk) = e(mk)

for all k ∈ N, i.e., let uk be a minimizer of the whole space problem with mass mk . Weneed to prove thatm ∈ I. By Lemma 4.9 there exists a minimizer u ∈ Ã∞(m) such thatuk ⇀ u weakly in BV (R3) and uk → u strongly in L1(R3). In particular, there holds˜E∞(u) = e(m) and hence m ∈ I. ��

Finally, we establish a few further properties of e(m).

Lemma 4.11. Let λ+m and λ−m be the supremum and the infimum, respectively, of the

Lagrangemultipliers in (4.9),amongall generalizedminimizers of ˜E∞ withmassm > 0.Then the function e(m) has left and right derivatives at each m ∈ (0,∞), and

limh→0+

e(m + h) − e(m)

h= λ−

m ≤ λ+m = limh→0+

e(m) − e(m − h)

h. (4.21)

In particular, e is a.e. differentiable and e′(m) = λ−m = λ+m =: λm for a.e. m > 0.


Proof. First of all, note that for m ≤ m0, where m0 is defined in Sect. 2, the functione(m) = m f (m) is given via (2.19), and the statement of the lemma can be verified explic-itly.On the other hand, bydefinitionwehaveλ−

m ≤ λ+m . Fixm > m0 and let (u1, . . . , uN ),with ui = χFi , be a generalized minimizer of ˜E∞ with mass m. We first show that

λ−m ≥ lim sup

h→0+

e(m + h) − e(m)

hand λ+m ≤ lim inf

h→0+

e(m) − e(m − h)

h. (4.22)

Indeed, for h > 0 let uhi = χFhiwith Fh

i = (m+hm )1/3Fi , so that |Fh

i | = (m+hm )|Fi |. Since

(m+hm )1/3 = 1 + h

3m + o(h), we have

˜E∞(uhi ) = ˜E∞(ui ) +2h

3m

∫

∂Fiκ(x) (x · ν(x)) dH2(x)

+h

12πm

∫

∂Fi

∫

Fi

(x · ν(x))

|x − y| dy dH2(x) + o(h), (4.23)

where ν(x) is the outward unit normal to ∂Fi at point x . In view of the Euler–Lagrangeequation (4.9), we hence obtain

˜E∞(uhi ) − ˜E∞(ui ) = λh

3m

∫

∂Fi(x · ν(x)) dH2(x) = λ

(

|Fhi | − |Fi |

)

+ o(h), (4.24)

where λ is the Lagrange multiplier in (4.9). Passing to the limit as h → 0+, this gives

lim suph→0+

e(m + h) − e(m)

h≤ lim sup

h→0+

1

h

(

N∑

i=1

˜E∞(uhi ) −N

∑

i=1

˜E∞(ui ))

≤ λ. (4.25)

Since (4.25) holds for all generalizedminimizers, this yields the first inequality in (4.22).Following the same argument with h replaced by −h, and taking the limit as h → 0+,we obtain the second inequality in (4.22).

Now, by Lemma 4.8 the function e(m) is a.e. differentiable on (0,∞), and at thepoints of differentiability we have e′(m) = λ−

m = λ+m =: λm . Hence, for any h > 0 thereexists mh ∈ (m,m + h) such that e is differentiable at mh and

e(m + h) − e(m)

h≥ e′(mh) = λmh , (4.26)

so that

lim infh→0+

e(m + h) − e(m)

h≥ λ := lim inf

h→0+λmh . (4.27)

Let hk → 0+ be a sequence such that λmhk→ λ as k → ∞. If (uk1, . . . , u

kN ) are general-

izedminimizerswithmassmhk then byLemma4.9 they converge, up to a subsequence, toa generalized minimizer with massm. In view of Lemma 4.4, up to another subsequencewe also have that the boundaries of the components of the generalized minimizers withmassmhk converge strongly in C

2 to those of the limit generalized minimizer with massm. Therefore, by (4.9) we have that λ is the Lagrange multiplier associated with the limitminimizer. It then follows that λ ≥ λ−

m , so that recalling (4.22) and (4.27) we get


limh→0+

e(m + h) − e(m)

h= λ−

m . (4.28)

This is the first equality in (4.21). The last equality in (4.21) follows analogously bytaking the limit from the other side. ��Remark 4.12. From the proof of Lemma 4.11 it follows that λ±

m are in fact the maximumand the minimum (not only the supremum and the infimum) of the Lagrange multipliersin (4.9), i.e., that λ±

m are attained by some generalized minimizers with mass m.

Corollary 4.13. The function e(m) is Lipschitz continuous on [m0,∞) for any m0 > 0.

Proof. This follows from (4.21), noticing that for all m ≥ m0 there holds

− ∞ < infm′∈[m0,M]

λ−m′ ≤ λ−

m ≤ λ+m ≤ supm′∈[m0,M]

λ+m′ < +∞, (4.29)

where M > 0 is such that I ⊂ [0, M], and we used (4.9) together with the uniformregularity from Lemma 4.4 for the components of the generalized minimizers. ��

4.4. Proof of Theorem 3.2. In lieu of a complete characterization of the function f (m)

and the set I, we show that f (m) is continuous and attains its infimum on I.The next result follows directly from Theorem 4.5, Theorem 4.7 and [40, Theo-

rem 3.2].

Lemma 4.14. There exists a universal constant δ0 > 0 such that for any m ∈ (0,∞)

there exist N ≥ 1 and m1, . . . ,mN ∈ I such that mi ≥ min{δ0,m} for all i = 1, . . . , Nand

m =N

∑

i=1

mi and f (m) =N

∑

i=1

mi

mf (mi ). (4.30)

Theorem 3.2 is a corollary of the following result.

Theorem 4.15. The function f (m) is Lipschitz continuous on [m0,∞) for any m0 > 0.Furthermore, f (m) attains its minimum, i.e.,

I∗ :={

m∗ ∈ I : f (m∗) = infm∈I

f (m)

}

�= ∅. (4.31)

Furthermore, we have f (m) ≥ f ∗ for all m > 0 and

limm→0

f (m) = ∞, limm→∞ f (m) = f ∗, lim

m→∞ ‖ f ′‖L∞(m,∞) = 0. (4.32)

Proof. Since f (m) = e(m)/m, we have that f (m) is Lipschitz continuous by Corol-lary 4.13. By the continuity of f (m) and since I is compact, it then follows that thereexists a (possibly non-unique) minimizer m∗ > 0 of f (m) over I. On the other hand,since f (m∗) ≤ f (m) for all m ∈ I, by Lemma 4.14 we obtain

f (m) =N

∑

i=1

mi

mf (mi ) ≥

N∑

i=1

mi

mf (m∗) = f (m∗) ∀m > 0. (4.33)


Turning to (4.32), the first statement there follows from (2.19). Let now u∗ = χF∗ ∈A∞(m∗) be a minimizer of ˜E∞ with m = m∗ for some m∗ ∈ I∗. Given k ∈ N, wecan consider k copies of F∗ sufficiently far apart as a test configuration. We hence getf (km∗) ≤ f (m∗) for any k ∈ N. Combining this with (4.33) yields the second identityin (4.32), once we establish the third identity. For the latter, by Corollary 4.13 and (2.20)we have

ess limm→∞ | f ′(m)| = ess lim

m→∞

∣

∣

∣

∣

e′(m)m − e(m)

m2

∣

∣

∣

∣

≤ ess limm→∞

f (m) + |e′(m)|m

= 0, (4.34)

which yields the claim. ��

5. Proof of Theorems 3.3 and 3.5

5.1. Compactness and lower bound. In this section, we present the proof of the lowerbound part of the �-limit in Theorem 3.3:

Proposition 5.1 (Compactness and lower bound). Let (uε) ∈ Aε, let με be given by(2.4), let vε be given by (2.7) and suppose that

lim supε→0

ε−4/3Eε(uε) < ∞. (5.1)

Then the following holds:(i) There exists μ ∈ M+(T)∩H′ and v ∈ H such that upon extraction of subsequences

we have με ⇀ μ inM(T) and vε ⇀ v inH. Furthermore,

− �v = μ − λ in D′(T). (5.2)

(ii) The limit measure satisfies

E0(μ) ≤ lim infε→0

ε−4/3Eε(uε). (5.3)

Proof. The proof proceeds via a sequence of 4 steps.Step 1: Compactness Since

∫

Tdμε = λ, it follows that there is μ ∈ M+(T) with

∫

Tdμ = λ and a subsequence such that με ⇀ μ inM(T). Furthermore, from (5.1) we

have the uniform bound

1

2

∫

T

|∇vε|2 dx = 1

2

∫

T

∫

T

G(x − y) dμε(x) dμε(y) ≤ ε−4/3Eε(uε) ≤ C. (5.4)

By the definition of the potential, we also have∫

Tvε dx = 0. Upon extraction of a

further subsequence, we hence get vε ⇀ v inH. Since με ⇀ μ inM(T) and since theconvolution of G with a continuous function is again continuous, we also have∫

T

(∫

T

G(x − y)ϕ(x) dx

)

dμε(y) →∫

T

(∫

T

G(x − y)ϕ(x) dx

)

dμ(y) ∀ϕ ∈ D(T).

(5.5)

This yields by Fubini–Tonelli theorem and uniqueness of the distributional limit that

v(x) =∫

G(x − y) dμ(y) for a.e. x ∈ T. (5.6)


Furthermore, since vε satisfies

− �vε = με − λ in D′(T), (5.7)

taking the distributional limit, it follows that v satisfies (5.2). In particular, (5.2) impliesthat μ defines a bounded functional on H, i.e. μ ∈ H′.Step 2: Decomposition of the energy into near field and far field contributions We splitthe nonlocal interaction into a far-field and a near-field component. For ρ ∈ (0, 1) andx ∈ T, let ηρ(x) := η(|x |/ρ), where η ∈ C∞(R) is a monotonically increasing functionsuch that η(t) = 0 for t ≤ 1

2 and η(t) = 1 for t ≥ 1. The far-field part Gρ and thenear-field part Hρ of the kernel G are then given by

Gρ(x) = ηρ(x)G(x), Hρ := G − Gρ. (5.8)

For any u ∈ Aε, we decompose the energy accordingly as Eε = E (1)ε + E (2)

ε , where

ε−4/3E (1)ε (u) = 1

2ε−4/3

∫

T

∫

T

Gρ(x − y)u(x)u(y) dx dy

ε−4/3E (2)ε (u) = ε−1/3

∫

T

|∇u| dx +1

2ε−4/3

∫

T

∫

T

Hρ(x − y)u(x)u(y) dx dy(5.9)

In the rescaled variables, the far field part E (1)ε of the energy can also be expressed as

ε−4/3E (1)ε (u) = 1

2

∫

T

∫

T

Gρ(x − y) dμε(x) dμε(y), (5.10)

where με is given by (2.4). For the near field part E(2)ε of the energy, we set �ε := ε−1/3

and define u : T�ε → R by

u(x) := u(x/�ε), (5.11)

where T�ε is a torus with sidelength �ε (cf. Sect. 2). In the rescaled variables, we get

ε−4/3E (2)ε (u) = ε1/3

(

∫

T�ε

|∇u|dx +1

2

∫

T�ε

∫

T�ε

ε1/3Hρ(ε1/3(x − y))u(x)u(y) dx dy

)

.

(5.12)

Step 3: Passage to the limit: the near field part Our strategy for the proof of the lowerbound for (5.12) is to compare E (2)

ε with the whole space energy treated in Sect. 4 anduse the results of this section. We claim that

lim infε→0

ε−4/3E (2)ε (u) ≥ (1 − cρ)λ f ∗, (5.13)

for some universal constant c > 0.Let �(x) := 1

4π |x | , x ∈ R3, be the Newtonian potential in R

3 and let �#(x) := 14π |x | ,

x ∈ T, be the restriction of �(x) to the unit torus. We also define the correspondingtruncated Newtonian potential �#

ρ : T → R by

�#ρ(x) := (1 − ηρ(x))�#(x). (5.14)


By a standard result, we have

G(x) = �#(x) + R(x), x ∈ T, (5.15)

for some R ∈ Lip(T). Hence

Hρ(x) = (1 − ηρ(x))G(x) ≥ (1 − ηρ(x))(�#(x) − ‖R‖L∞(T))

≥ (1 − ηρ(x))�#(x)(1 − 4πρ‖R‖L∞(T)) = (1 − cρ)�#ρ(x), (5.16)

where c = 4π‖R‖L∞(T). Inserting this estimate into (5.12), for cρ < 1 we arrive at

ε−4/3E (2)ε (u)

1 − cρ≥ ε1/3

(

∫

T�ε

|∇u| dx +1

2

∫

T�ε

∫

T�ε

ε1/3�#ρ(ε1/3(x − y))u(x)u(y) dx dy

)

= ε1/3

(

∫

T�ε

|∇u| dx +∫

T�ε

∫

T�ε

(1 − ηρ(ε1/3(x − y)))

8π |x − y| u(x)u(y) dx dy

)

.

(5.17)

Next we want to pass to a whole space situation by extending the function u period-ically to the whole of R

3 and then truncating it by zero outside one period. We claimthat after a suitable translation there is no concentration of the periodic extension of u,still denoted by u for simplicity, on the boundary of a cube Q�ε := (− 1

2�ε,12�ε)

3. Moreprecisely, we claim that

∫

∂Q�ε

u(x − x∗) dH2(x) ≤ 6λ, (5.18)

for some x∗ ∈ Q�ε . Indeed, by Fubini’s theorem we have

λ�ε =∫

Q�ε

u dx =∫ 1

2 �ε

− 12 �ε

H2({u(x) = 1} ∩ {x · e1 = t}) dt, (5.19)

where e1 is the unit vector in the first coordinate direction. This yields existence ofx∗1 ∈ (− 1

2�ε,12�ε) such that H2({u(x) = 1} ∩ {x · e1 = x∗

1 }) ≤ λ. Repeating thisargument in the other two coordinate directions and taking advantage of periodicity ofu, we obtain existence of x∗ ∈ Q�ε such that (5.18) holds.

Now we set

u(x) :={

u(x − x∗) x ∈ Q�ε,

0 x ∈ R3\Q�ε

.(5.20)

We also introduce the truncated Newtonian potential on R3 by

�ρ(x) := 1 − ηρ(x)

4π |x | , x ∈ R3. (5.21)

By (5.18), the additional interfacial energy due to the extension (5.20) is controlled:∫

T�ε

|∇u| dx =∫

R3|∇u| dx −

∫

∂Q�ε

u dx ≥∫

R3|∇u| dx − 6λ. (5.22)


We hence get from (5.17):

ε−4/3E (2)ε (u)

1 − cρ≥ ε1/3

(∫

R3|∇u| dx +

1

2

∫

R3

∫

R3�ε−1/3ρ(x − y)u(x)u(y) dx dy − 6λ

)

≥ λ∫

R3 u dx

(∫

R3|∇u| dx +

1

2

∫

R3

∫

R3�ρ0(x − y)u(x)u(y) dx dy

)

− 6λε1/3,

(5.23)

for any ρ0 > 0, provided that ε is sufficiently small (depending on ρ0). By Corollary 4.6and Theorem 4.15, the first term on the right hand side is bounded below by λ f ∗ as soonas ρ0 ≥ R1. Therefore, passing to the limit as ε → 0, we obtain (5.13).Step 4: Passage to the limit: the far field part Passing to the limit με ⇀ μ inM(T), forthe far field part of the energy we obtain

limε→0

ε−4/3E (1)ε (uε) = 1

2

∫

T

∫

T

Gρ(x − y) dμ(x) dμ(y). (5.24)

At the same time, by (A.13) in Lemma A.2 in the appendix the set {(x, y) ∈ T : x = y}is negligible with respect to the product measure μ ⊗ μ on T × T. Therefore, sinceGρ(x − y) ↗ G(x − y) as ρ → 0 for all x �= y, by the monotone convergence theoremthe right-hand side of (5.24) converges to

∫

T

∫

TG(x−y) dμ(x) dμ(y). Finally, the lower

bound in (5.3) is recovered by combining this result with the limit of (5.13) asρ → 0. ��

5.2. Upper bound construction. We next give the proof of the upper bound in Theo-rem 3.3:

Proposition 5.2 (Upper bound construction). For anyμ ∈ M+(T)∩H′ with∫

Tdμ = λ,

there exists a sequence (uε) ∈ Aε such that

με ⇀ μ inM(T) and vε ⇀ v inH, (5.25)

as ε → 0, where με, vε and v are defined in (2.4), (2.7) and (3.4), respectively, and

lim supε→0

ε−4/3Eε(uε) ≤ E0(μ). (5.26)

Proof. We first note that the limit energy is continuous with respect to convolutions. Inparticular, we may assume without loss of generality that dμ(x) = g(x)dx for someg ∈ C∞(T), and that there exist C ≥ c > 0 such that

c ≤ g(x) ≤ C for all x ∈ T. (5.27)

We proceed now to the construction of the recovery sequence. For δ > 0, we parti-tion T into cubes Qδ

i with sidelength δ. Let u∗ ∈ BV (T�ε ; {0, 1}), where �ε = ε−1/3,be a minimizer of ˜E∞ over Ã∞(m) with m = m∗ ∈ I∗ (cf. Theorem 4.15), suitablytranslated, restricted to a cube with sidelength �ε and then trivially extended to T�ε (thelatter is possible without modifying either the mass or the perimeter by Theorem 4.7 for


universally small ε). For a given set of centers a( j)ε,δ , j = 1, . . . , Nε,δ , and a given set of

scaling factors θ( j)ε,δ ∈ [1,∞), we define uε,δ : T → R by

uε,δ(x) :=Nε,δ∑

j=1

u∗ (

θ( j)ε,δ ε−1/3(x − a( j)

ε,δ ))

for x ∈ T, (5.28)

as the sum of Nε,δ suitably rescaled minimizers of ˜E∞(u)/∫

R3 u dx . Note that∫

T�εu∗(ε−1/3x) dx = εm∗. To decide on the placement of a( j)

ε,δ , we denote the number

of the centers in each cube as N (i)ε,δ , i.e.,

N (i)ε,δ := #

{

j ∈ {1, . . . , Nε,δ} : a( j)ε,δ ∈ Qδ

i

}

. (5.29)

With this notation we have Nε,δ = ∑

i N(i)ε,δ , provided that supp(uε,δ) ∩ ∂Qδ

i = ∅

for all i . The measure μ is then locally approximated in every cube Qδi by “droplets”

uniformly distributed throughout each cube. Namely, we set

N (i)ε,δ =

⌈

μ(Qδi )

ε1/3m∗

⌉

, (5.30)

and choose a( j)ε,δ so that

K ε1/9 ≤ dε,δ ≤ K ′ε1/9, (5.31)

where dε,δ := mini �= j |a( j)ε,δ −a(i)

ε,δ| is the minimal distance between the centers, for someK ′ > K > 0 depending only on μ. We also set

θ( j)ε,δ :=

(

ε1/3m∗N (i)ε,δ

μ(Qδi )

)1/3

if a( j)ε,δ ∈ Qδ

i . (5.32)

Then, if ε is sufficiently small depending only on δ and μ, we find that uε,δ ∈ Aε.Finally, we define the measure με,δ associated with the test function uε,δ constructed

above, dμε,δ(x) := ε−2/3uε,δ(x) dx , as in (2.4) and choose a sequence of δ → 0. Choos-ing a suitable sequence of ε = εδ → 0, we have μεδ,δ ⇀ μ inM(T). For simplicity ofnotation, in the following we will suppress the δ-dependence, e.g., we will simply writeuε instead of uεδ,δ , etc.

It remains to prove (5.26). As in the proof of the lower bound, for a given ρ ∈ (0, 1)we split the kernel G into the far field part Gρ and the near field part Hρ . Decomposingthe energy into the two parts in (5.9) and using (5.10), we have

ε−4/3E (1)ε (uε) = 1

2

∫

T

∫

T

Gρ(x − y) dμε(x) dμε(y). (5.33)

Since με ⇀ μ inM(T), we can pass to the limit ε → 0 in (5.33). Then, since the limitmeasure μ belongs to H′, by the monotone convergence theorem we recover the fullCoulombic part of the limit energy E0 in (3.1) in the limit ρ → 0.


For the estimate of the near field part of the energy, we observe that

Hρ(x) ≤ (1 + cρ)�#ρ(x), (5.34)

for some universal c > 0 (cf. the estimates in (5.16)). With this estimate, we get

ε−4/3E (2)ε (uε) ≤ ε−1/3

∫

T

|∇uε| dx +1

2ε−4/3(1 + cρ)

∫

T

∫

T

�#ρ(x − y)uε(x)uε(y) dx dy

≤ ε−1/3∫

T

|∇uε| dx + ε−4/3(1 + cρ)

∫

T

∫

B 12 dε

(x)

uε(x)uε(y)

8π |x − y| dy dx

+ ε−4/3(1 + cρ)

∫

T

∫

Bρ(x)\B 12 dε

(x)

uε(x)uε(y)

8π |x − y| dy dx . (5.35)

By the optimality of u∗ and the fact that all θ( j)ε,δ ≥ 1, we hence get

ε−1/3∫

T

|∇uε| dx + (1 + cρ)ε−4/3∫

T

∫

B 12 dε

(x)

uε(x)uε(y)

8π |x − y| dy dx

≤ (1 + cρ)(λ + oε(1)) f∗, (5.36)

where the oε(1) term can be made to vanish in the limit by choosing εδ small enough foreach δ to ensure that all θ

( j)ε,δ → 1. Since we can choose ρ > 0 arbitrary, this recovers

the first term in the limit energy E0 in (3.1).It hence remains to estimate the last term in (5.35). We first note that

ε−4/3∫

T

∫

Bρ(x)\B 12 dε

(x)

uε(x)uε(y)

|x − y| dy dx ≤ λ supx∈T

∫

Bρ(x)\B 12 dε

(x)

dμε(y)

|x − y| . (5.37)

To control the last term, for any given x ∈ T we introduce a family of dyadic ballsBk := B2−kρ(x), k = 0, 1, . . . . By (5.31), we have Bρ(x)\B 1

2 dε(x) ⊂ ⋃Kε

k=0 Bk\Bk+1

for Kε := �log2(ρ/dε)� ≤ 1 + log2(ρ/dε), or, equivalently, 2−Kερ ≥ dε

2 , provided thatε is sufficiently small depending only on δ and μ. Therefore, with our construction wehave με(Bk) ≤ 2−3kCρ3 for some C > 0 depending only on μ and all 0 ≤ k ≤ Kε.This yields

supx∈T

∫

Bρ(x)\B 12 dε

(x)

dμε(y)

|x − y| ≤Kε∑

k=0

∫

Bk\Bk+1dμε(y)

|x − y|

≤Kε∑

k=0

2k+1με(Bk)

ρ≤

Kε∑

k=0

2Cρ2

4k≤ 8Cρ2

3. (5.38)

Since we can choose ρ > 0 arbitrarily small, this concludes the proof. ��Remark 5.3. We note that the construction in Proposition 5.2 still yields, upon extractionof a subsequence, a recovery sequence for a given sequence of ε = εn → 0.


5.3. Equidistribution of energy. We now prove Theorem 3.5. First, we observe that

dνε = ε−1/3|∇uε| dx +1

2vεdμε, (5.39)

where με is defined in (2.4). We claim that the following lower bound for measures νε,given x ∈ T and δ ∈ (0, 1), holds true:

lim infε→0

νε(Bδ(x)) ≥ |Bδ(x)|λ f ∗. (5.40)

As in (5.8), we split G into the far field part Gρ and the near field part Hρ , for somefixed ρ ∈ (0, δ). Since supp(Hρ) ⊂ Bδ(0), we obtain

νε(Bδ(x)) = ε−1/3∫

Bδ(x)|∇uε| dx +

1

2ε−4/3

∫

Bδ(x)

∫

Bδ(x)Hρ(x − y)uε(x)uε(y) dy dx

+1

2

∫

Bδ(x)

∫

T

Gρ(x − y) dμε(y) dμε(x). (5.41)

Then, since Gρ is smooth and με(T) = λ, by Corollary 3.4 the integral∫

TGρ(x −

y) dμε(y) converges toλ∫

TGρ(y) dy uniformly in x ∈ T as ε → 0.At the same time, by

the definition ofG and (5.34) we have 0 = ∫

TG(y) dy = ∫

TGρ(y) dy+

∫

THρ(y) dy ≤

∫

TGρ(y) dy + Cρ2 for some universal C > 0. Hence, we get

νε(Bδ(x)) ≥ ε−1/3∫

Bδ(x)|∇uε| dx

+1

2ε−4/3

∫

Bδ(x)

∫

Bδ(x)Hρ(x − y)uε(x)uε(y) dy dx − Cλρ2, (5.42)

for ε sufficiently small and C > 0 universal.We now identify uε with its periodic extension to the whole of R

3. By Fubini’stheorem, for a given δ′ ∈ (0, δ), there is t = tδ′,δ ∈ (δ′, δ) such that

∫

∂Bt (x)uε(x) dH2(x) ≤ 1

δ − δ′

∫ δ

δ′

(∫

∂Bs (x)uε(x) dH2(x)

)

ds

= 1

δ − δ′

∫

Bδ(x)\Bδ′ (x)uε dx . (5.43)

We then define uε ∈ BV (R3; {0, 1}) by uε = uεχBt (x). Recalling again Corollary 3.4,we obtain

∫

R3|∇uε| dx =

∫

Bt (x)|∇uε| dx +

∫

∂Bt (x)uε(x) dH2(x)

≤∫

Bδ(x)|∇uε| dx + Cλδ2ε2/3, (5.44)

for some universal C > 0, provided that ε is sufficiently small. We note that uε(x) ≤uε(x) for every x ∈ R

3. Furthermore, for sufficiently small δ we have Hρ ≥ 0 and

Hρ(x − y) ≥ (1 − cρ)�(x − y) for all |x − y| ≤ 12ρ, (5.45)


for some universal c > 0 (where � is the Newtonian potential in R3, as above). From

(5.42), (5.44) and (5.45) we then get

νε(Bδ(x)) ≥ ε−1/3∫

R3|∇uε| dx +

1 − cρ

2ε−4/3

×∫

R3

∫

Bρ/2(x)�(x − y)uε(x)uε(y) dy dx − Cλρ2, (5.46)

for ε small enough. Letting now uε(x) := uε(ε1/3x) be the rescaled function which

satisfies∫

R3uε dx = 1

ε

∫

R3uε dx = λ|Bt (x)|ε−1/3 + o(ε−1/3), (5.47)

for every fixed ρ0 > 0 and ε sufficiently small, we get

νε(Bδ(x))

≥ (1 − cρ)ε1/3

(

∫

R3|∇uε| dx +

1

2

∫

R3

∫

Bρ0 (x)�(x − y)uε(x)uε(y) dy dx

)

− Cλρ2

≥ (1 − 2cρ)λ|Bt (x)|∫

R3 uε dx

(∫

R3|∇uε| dx +

1

2

∫

R3

∫

R3�ρ0 (x − y)uε(x)uε(y) dy dx

)

− Cλρ2,

(5.48)

where�ρ0 is defined via (5.21). RecallingCorollary 4.6 and choosingρ0 ≥ R1,we obtain

lim infε→0

νε(Bδ(x)) ≥ (1 − 2cρ)λ f ∗|Bt (x)| − Cλρ2, (5.49)

which gives (5.40) by first letting ρ → 0 and then δ′ → δ.We now prove a matching upper bound. Notice that by the definition we have

vε(x) ≥ C := −λ|miny∈T G(y)| for every x ∈ T. Therefore, the negative part ν−ε

of νε obeys ν−ε (U ) = − 1

2

∫

U∩{vε<0} vεdμε ≤ 12 |C |με(U ) for every open set U ⊂ T.

In turn, since νε(T) = λ f ∗ + oε(1) by (3.8), it follows that the positive part ν+ε of ν

obeys ν+ε (U ) = ∫

U∩{vε≥0}(

ε−1/3|∇uε| dx + 12vε dμε

) ≤ λ f ∗ + 12 |C |λ + oε(1). Hence

|νε| = ν+ε + ν−ε is uniformly bounded as ε → 0, and up to a subsequence νε ⇀ ν

for some ν ∈ M(T) with ν(T) = λ f ∗. Since from the lower bound (5.40) we haveν(U ) ≥ λ f ∗|U |, it then follows that dν = λ f ∗dx . Finally, in view of the uniqueness ofthe limit measure, the result holds for the original sequence of ε → 0. ��

6. Uniform Estimates for Minimizers of the Rescaled Energy

In this section, we establish uniform estimates for the minimizers of the rescaled prob-lem associated with ˜E� over Ã� from (2.23) and (2.24), respectively. The main result isa uniform bound on the modulus of the potential, independently of the domain size �.

Throughout this section, F ⊂ T� with |F | = λ� is always taken to be such thatu� = χF is a regular representative of a minimizer of ˜E� over Ã� for a given λ > 0 (forsimplicity of notation, we suppress the explicit dependence of F on � throughout thissection). The estimates below are obtained for families of minimizers (u�n ) as �n → ∞


and hold for all �n ≥ �0, where �0 > 0 may depend on λ and the choice of the family.For simplicity of notation, we indicate this by saying that an estimate holds for � � 1.

Following [46,61] we recall the notion of (�, r0)-minimizer of the perimeter (for adifferent approach that leads to the same regularity results, see [34]).

Definition 6.1. Given �, r0 > 0 we say that a set F ⊂ T� is a volume-constrained(�, r0)-minimizer if

P(F) ≤ P(F ′) + �|F�F ′| ∀ F ′ ⊂ T�, s.t. (F�F ′) ⊂ Br0 and |F ′| = |F |, (6.1)

where P(F) denotes the perimeter of the set F , and Br denotes a generic ball of radiusr contained in T�.

The following result is a consequence of the regularity theory for minimal surfaceswith volume constraint (see for instance [46, Chapters III–IV], [61, Section 4]).

Proposition 6.2. Let F ⊂ T� be a volume-constrained (�, r0)-minimizer, with |F | ∈(

r30 , �3 − r30

)

. Then ∂F is of class C1,α and there exist universal constants δ > 0 andc > 0 such that for all x0 ∈ F we have

|F0 ∩ Br (x0)| ≥ cr3 for all r ≤ min

(

r0,δ

�

)

, (6.2)

where F0 is the connected component of F such that x0 ∈ F0.

Let �(x) := 14π |x | , x ∈ R

3, be the Newtonian potential in R3 and let �#

� (x) :=1

4π |x | , x ∈ T�, be the restriction of �(x) to T�. Setting

˜G�(x) := 1

�G

( x

�

)

, x ∈ T�, (6.3)

by (5.15) we have for all � ≥ 1:

˜G�(x) = �#� (x) + R�(x) for all x ∈ T�, (6.4)

with R� ∈ Lip(T�) satisfying

|R�(x)| ≤ C

�and |∇R�(x)| ≤ C

�2for all x ∈ T�, (6.5)

with a universal C > 0. Let now

vF (x) :=∫

F

˜G�(x − y) dy, x ∈ T�, (6.6)

be the potential associated with F . Notice that vF satisfies

− �vF = χF − λ

�2and

∫

T�

vF dx = 0. (6.7)

In particular, by standard elliptic regularity we have vF ∈ C1,α(T�) for any α ∈ (0, 1)[31], and vF is subharmonic outside F , so that the maximum of vF is attained in F . Themain result concerning vF that enables our uniform estimates for minimizers χF of ˜E�

is uniform boundedness of vF in T� which is contained in Theorem 6.9. To achieve thisbound, we prove several auxiliary results for vF in the spirit of potential theory. Theseresults are contained in Lemmas 6.3, 6.5 and 6.7 that follow.

We begin with the following a priori bound for vF (throughout the rest of this section,vF always refers to the potential associated with the minimizer F).


Lemma 6.3. There exists a universal constant C > 0 such that

− C ≤ vF ≤ C(λ�)2/3, (6.8)

for all � � 1.

Proof. First of all, observe that vF (x) = vε(ε1/3x) for ε = �−3, where vε is defined

in (2.7), in which με is given by (2.4) with uε(x) = χF (ε−1/3x). Furthermore, by arescaling we have that uε is a minimizer of Eε over Aε. Therefore, to establish a lowerbound for vF , it is sufficient to do so for vε.

Let Gρ and Hρ be as in (5.8) (with the choice of η fixed once and for all), and notethat there exists a universal ρ0 > 0 such that Hρ ≥ 0 for all ρ ∈ (0, ρ0) and, hence,

vε(x) ≥∫

T

Gρ(x − y) dμε(y). (6.9)

At the same time, by Corollary 3.4 and boundedness of |∇Gρ | we have∫

T

Gρ(x − y) dμε(y) → λ

∫

T

Gρ(y) dy uniformly in x ∈ T, (6.10)

as ε → 0. Notice that from the definition of G we have 0 = ∫

TG(x) dx =

∫

TGρ(x) dx +

∫

THρ(x) dx . Therefore, by (5.15) we get

− Cρ2 ≤∫

T

Gρ(x) dx ≤ 0, (6.11)

for some universal C > 0 and all ρ ∈ (0, ρ0). Choosing ρ = min{ρ0, λ−1/2}, we thenobtain vε ≥ −2C for all ε > 0 sufficiently small.

On the other hand, by (6.5) there exists a universal constant C > 0 such that

vF (x) ≤ C∫

F

dy

|x − y| ≤ C

(∫

BR(x)

dy

|x − y| +|F\BR(x)|

R

)

≤ C(2πR2 + R−1|F |), (6.12)

for any � ≥ 1 and R > 0. The claim then follows by choosing R = |F |1/3 = (λ�)1/3.��

Remark 6.4. Let λ0 > 0 and let λ ∈ (0, λ0). Since vF ≥ λminx∈T G(x), it is alsopossible to obtain a lower bound on vF which depends only on λ0, and not on the familyof the minimizers, provided that � ≥ �0 for some �0 > 0 depending only on λ. In thiscase all the estimates of this section still hold, but with constants that depend on λ0.

We next obtain a pointwise estimate of the gradient of vF in terms of vF itself.

Lemma 6.5. There exists a universal constant C > 0 such that for every � � 1 we have

|∇vF (x)| ≤ 3

2(vF (x) + C) , (6.13)

for any x ∈ T�.


Proof. Without loss of generality we may assume that x = 0. Arguing as in the proofof Lemma 6.3 and with the same notation, we can write

|∇vF (0)| ≤∫

F|∇˜G�(y)| dy = �

∫

T

|∇G(y)| χF (y�) dy = ε1/3∫

T

|∇G(y)| dμε(y)

≤ ε1/3∫

T

|∇Gρ(y)| dμε(y) + ε1/3∫

T

|∇Hρ(y)| dμε(y), (6.14)

where we recalled that ε = �−3. Using (5.15), we have

|∇Hρ(y)| ≤ (1 + cρ)|y|−1Hρ(y) + C |y|−1ρ−1χBρ\Bρ/2(y), (6.15)

for someuniversal c,C > 0 and allρ ∈ (0, ρ0). Substituting this into (6.14) and recalling(2.4) and (5.34), we obtain

|∇vF (0)|1 + cρ

≤∫

T\Bε1/3 (0)

Hρ(y) dμε(y) + ε−1/3∫

Bε1/3 (0)

|y|−1Hρ(y) uε(y) dy

+Cε1/3ρ−1∫

Bρ(0)\Bρ/2(0)|y|−1 dμε(y) + ε1/3

∫

T

|∇Gρ(y)| dμε(y)

≤∫

T

Hρ(y) dμε(y) + C ′(1 + ε1/3ρ−2λ) + ε1/3∫

T

|∇Gρ(y)| dμε(y),

(6.16)

for some universal C,C ′ > 0. Since by Corollary 3.4 and the smoothness of Gρ wehave

∫

T|∇Gρ(x − y)| dμε(y) → λ

∫

T|∇Gρ(y)| dy uniformly in x ∈ T as ε → 0, it

is possible to choose ε0 > 0 sufficiently small independently of x such that the last twoterms in the right-hand side of (6.16) are bounded by a universal constant for all ε < ε0.Possibly reducing the value of ε0, for every ρ ≤ 1/(2c) and ε < ε0, with ε0 dependingon ρ, we also have

2

3|∇vF (0)| ≤ vF (0) + C −

∫

T

Gρ(y) dμε(y), (6.17)

where we took into account that vF (0) = ∫

TGρ(y) dμε(y)+

∫

THρ(y) dμε(y). Finally,

using (6.10) and (6.11), we obtain

|∇vF (0)| ≤ 3

2vF (0) + C(1 + λρ2), (6.18)

for some universal C > 0 and all ε < ε0, again, possibly decreasing the value of ε0.The proof is concluded by choosing ρ ≤ λ−1/2. ��Corollary 6.6. Let � � 1 and let x ∈ F be a global maximum of vF . Then

vF (y) ≥ 3

4vF (x) − 1

4C for all y ∈ B1/6(x), (6.19)

where C is as in (6.13). Furthermore, if∫

Br (x0)vF (x) dx ≤ C ′|Br | for some x0 ∈ T�,

r ≤ 16 and C ′ > 0, then


vF (y) ≤ C + 2C ′ for all y ∈ Br (x0), (6.20)

Proof. Since vF ∈ C1(T�), for any y ∈ B1/6(x) there exists θ ∈ (0, 1) such that withthe help of (6.13) we have

vF (x) − vF (y) = ∇vF (θ x + (1 − θ)y) · (x − y)

≤ 1

6|∇vF (θ x + (1 − θ)y)|

≤ 1

4vF (θ x + (1 − θ)y) +

1

4C

≤ 1

4vF (x) +

1

4C. (6.21)

Similarly, letting y be a global maximum of vF in Br (x0) and letting x1 ∈ Br (x0)be such that vF (x1) = |Br |−1

∫

Br (x0)vF (x) dx , we may write

vF (y) ≤ vF (y) − vF (x1) + C ′

≤ |∇vF (θx1 + (1 − θ)y)| |y − x1| + C ′

≤ 1

2vF (θ x1 + (1 − θ)y) +

1

2C + C ′

≤ 1

2vF (y) +

1

2C + C ′, (6.22)

which completes the proof. ��The next lemma provides a basic estimate for the variation of the Coulombic energy

under uniformly bounded perturbations.

Lemma 6.7. There exists a universal constant C > 0 such that for any � ≥ 1 and forany F ′ ⊂ T�, with F�F ′ ⊂ Br (x0) for some x0 ∈ T� and r > 0, there holds

∣

∣

∣

∣

∫

FvF dx −

∫

F ′vF ′ dx

∣

∣

∣

∣

≤(

2‖vF‖L∞(T�) + Cr2)

|F�F ′|. (6.23)

Proof. By direct computation, we have∣

∣

∣

∣

∫

FvF dx −

∫

F ′vF ′ dx

∣

∣

∣

∣

=∣

∣

∣

∣

∫

T�

∫

T�

(

χF (x)˜G�(x − y)χF (y) − χF ′(x)˜G�(x − y)χF ′(y))

dx dy

∣

∣

∣

∣

=∣

∣

∣

∣

∫

T�

∫

T�

(χF (x) + χF ′(x))˜G�(x − y)(χF (y) − χF ′(y)) dx dy

∣

∣

∣

∣

≤ 2

∣

∣

∣

∣

∫

T�

∫

T�

χF (x)˜G�(x − y)(χF (y) − χF ′(y)) dx dy

∣

∣

∣

∣

+

∣

∣

∣

∣

∫

T�

∫

T�

(χF (x) − χF ′(x))˜G�(x − y)(χF (y) − χF ′(y)) dx dy

∣

∣

∣

∣

≤ 2∫

T�

|vF (y)| |χF (y) − χF ′(y)| dy


+ 2∫

T�

∫

Br (y)|˜G�(x − y)| |χF (y) − χF ′(y)| dx dy

≤(

2‖vF‖L∞(T�) + Cr2)

|F�F ′|, (6.24)

for some universal C > 0, where we used (6.4) and (6.5) in the last line. ��Lemma 6.7 implies that minimizers of ˜E� are volume constrained (�, r0)-minimizers

of the perimeter for r0 = 1 and � = C(‖vF‖L∞(T�) + 1), with C > 0 universal, when-ever � ≥ 1. In particular, by Lemma 6.3 we get � ≤ C(λ�)2/3, provided that � � 1.Therefore, from Proposition 6.2 we obtain the following result.

Proposition 6.8. There exist universal constants c > 0 and δ > 0 such that for all � � 1and all x0 ∈ F there holds

|F0 ∩ Br (x0)| ≥ cr3 for all r ≤ δ

(λ�)2/3, (6.25)


We now show that the potential vF is bounded in L∞(T�) by a universal constant as� → ∞.

Theorem 6.9 (L∞-estimate on the potential). There exists a universal constant C > 0and a constant �0 > 0 such that for all � ≥ �0 we have

‖vF‖L∞(T�) ≤ C. (6.26)

Proof. Observe first that by (6.8) we have vF ≥ −C , for some universal constantC > 0and � � 1. Therefore, letting V := max

x∈T�

vF (x), the thesis is equivalent to showing that

V ≤ C, (6.27)

for some universal C > 0 and large enough �.We first prove (6.27) with the constant depending only on λ. Partition T� into N

cubes of sidelength L = � N−1/3, with N 1/3 chosen to be the smallest integer such thatL ≤ min

( 16c

1/3λ−1δ, 13

)

, where c and δ are as in (6.25). Note that with our choice of Lwe have N ≥ 216λ3�3/(cδ3). If � is sufficiently large (depending on λ), we also havethat δ(λ�)−2/3 ≤ 1

2 L ≤ 112c

1/3λ−1δ. In particular, any ball of radius δ(λ�)−2/3 can beinscribed into a union of 27 adjacent cubes of the partition and stay at least distanceδ(λ�)−2/3 from the boundary of that union. Hence, by (6.25) and a counting argumentwe get that at least 7

8N cubes do not intersect F , so that we can find disjoint ballsB1, . . . , BM of radius 1

2 L ≤ 16 not intersecting F , with M ≥ 7

8N .Recalling that

∫

T�vF dx = 0 and that vF is bounded below by−C , for � � 1 we get

0 =∫

T�

vF dx ≥M

∑

i=1

∫

BivF dx − C�3. (6.28)

It follows that there exists an index i such that, for some universal C ′ > 0, we have∫

BivF dx ≤ CM−1�3 ≤ C ′|Bi |. (6.29)


We then apply the second part of Corollary 6.6 with x0 = xi , where xi is the center ofBi , to obtain

|vF (x)| ≤ C for all x ∈ Bi , (6.30)

for some universal C > 0.Let now x ∈ F be a global maximum of vF , so that vF (x) = V , and assume that

H2(F ∩ ∂Br (x)) ≥ 1

9V |F ∩ Br (x)| for any r ∈ (0, L/2). (6.31)

Letting m(r) := |F ∩ Br (x)|, so that dm(r)dr = H2(F ∩ ∂Br (x)) for a.e. r , (6.31) can be

written as

dm(r)

dr≥ 1

9Vm(r) for a.e. r ∈ (0, L/2). (6.32)

Integrating (6.32) over (r0, L/2), we get

m(r0) ≤ m(L/2) eV (r0−L/2)/9. (6.33)

Notice now that, as in Proposition 6.8, from Lemma 6.7 it follows that

m(r) ≥ cr3 for all r ≤ min

(

1,δ

V

)

. (6.34)

In particular, if r0 = δ/V ≤ L/4, we have

cδ3

V 3 ≤ m(r0) ≤ CL3e−LV/36, (6.35)

for some universal constant C > 0, which implies (6.27) with the constant dependingonly on λ.

On the other hand, if (6.31) does not hold, there exists r ∈ (0, L/2) such that

H2(F ∩ ∂Br (x)) <1

9V |F ∩ Br (x)|. (6.36)

We claim that, as in the proof of Lemma 4.2, if (6.27) does not hold, it is convenient tomove the set F∩Br (x) inside the ball Bi . Indeed, we define Fi := (xi − x)+(F∩Br (x))and u = u� −χF∩Br (x) +χFi . Note that by construction F ∩ Bi = ∅, so u is admissible.By minimality of u� and using (6.5), (6.30) and (6.36), we get

˜E�(u�) ≤ ˜E�(u)

= ˜E�(u�) + 2H2(F ∩ ∂Br (x)) +∫

FivF dx −

∫

F∩Br (x)vF dx

−∫

Fi

∫

F∩Br (x)

˜G�(x − y) dx dy +∫

F∩Br (x)

∫

F∩Br (x)

˜G�(x − y) dx dy

< ˜E�(u�) +

(

2

9V + C

)

|F ∩ Br (x)| −∫

F∩Br (x)vF dx, (6.37)


for some universal C > 0, provided that � � 1. Notice now that Corollary 6.6 impliesthat

vF (x) ≥ 3

4V − C for any x ∈ Br (x), (6.38)

for a universal C > 0. Hence

0 <

(

C − 1

2V

)

|F ∩ Br (x)|, (6.39)

for some universal C > 0 and � � 1, which leads to a contradiction if V is too large.Lastly, to establish (6.27) with C universal, we note that using (6.27) with the con-

stant depending on λ one gets that the density estimate in (6.25) holds for all r ≤ r0with some r0 > 0 depending only on λ, for � � 1. We can then repeat the coveringargument at the beginning of the proof with L > 0 universal, provided that � � 1, andobtain the conclusion by repeating the above argument. ��

From Theorem 6.9 and the arguments leading to Proposition 6.8, we obtain animproved density estimate for minimizers of ˜E�.

Corollary 6.10. There exist a universal constant c > 0 and a constant �0 > 0 such thatfor all x0 ∈ F and all � ≥ �0 we have

|F0 ∩ Br (x0)| ≥ cr3 for all r ≤ 1, (6.40)


Finally, we establish a uniform diameter bound for the connected components of theminimizers in Theorem 6.9.

Lemma 6.11 (Diameter bound). Let F0 be a connected component of F. Then thereexists a universal constant C > 0 such that

diam F0 ≤ C, (6.41)

for all � � 1.

Proof. Assume that diam F0 ≥ 2. Arguing as in the proof of Lemma 6.5 and using itsnotations, for any x ∈ T� and a universally small ρ0 > 0 we have

vF (x) ≥∫

F∩Bε−1/3ρ/2(x)

dy

8π |x − y| +∫

T

Gρ(ε1/3x − y) dμε(y), (6.42)

for all ρ ∈ (0, ρ0). Observe that by (6.10) and (6.11) the last term in the right-hand sideof (6.42) can be bounded below by −2Cλρ2, for � � 1 and C > 0 universal. Takingρ ≤ λ−1/2 and using (6.26), we then get

∫

F∩BR(x)

dy

|x − y| ≤ C, (6.43)

with a universal C > 0, for any R ≥ 1 and x ∈ T�, provided that � � 1 independentlyof x .

Recalling (6.40) and arguing as in Lemma 4.2, for all � � 1 there exists x0 ∈ F0such that

C ≥∫

F0∩BR(x0)

dy

|x0 − y| ≥ cmin{log (diam F0) , log R}, (6.44)

for some universal c,C > 0. The claim then follows by choosing a universal R that issufficiently large. ��


7. Proof of Theorem 3.6

For λ > 0, let (uε) ∈ Aε be a family of the regular representatives of minimizers of Eε,and let Nε and uε,k ∈ BV (R3; {0, 1}) be as in the statement of the theorem. Withoutloss of generality we may set xε,k = 0 in the statements below. We need to show thatthere exists ε0 > 0 such that for all ε ≤ ε0:

(i) There exist universal constants C, c > 0 such that

‖vε‖L∞(T�) ≤ C,

∫

R3uε,k(x) dx ≥ cε. (7.1)

(ii) There exist universal constants C, c > 0 such that

supp(uε,k) ⊆ BCε1/3(0), cλε−1/3 ≤ Nε ≤ Cλε−1/3. (7.2)

(iii) There exists a collection of indices Iε such that (#Iε)/Nε → 1 as ε → 0 and, uponextraction of a subsequence, for every sequence εn → 0 and every kn ∈ Iεn there

holds un → u in L1(R3), where un(x) := uεn ,kn (ε1/3n x), and u is a minimizer of

˜E∞ over Ã∞(m∗) for some m∗ ∈ I∗.The estimate for the potential in (i) follows from Theorem 6.9, setting u�ε = uε(·/�ε) ∈Ã�ε with �ε = ε−1/3 and noting that with u�ε = χF we have vF = vε(·/�ε). Similarly,the volume estimate in (i) follows from Corollary 6.10. The inclusion in (ii) followsfrom Lemma 6.11 by a rescaling. The estimate for Nε in (ii) follows from (i) and thefact that

∫

Tuε dx = λε2/3.

We turn to the proof of statement (iii). Given δ > 0, let Nε,δ ≥ 0 be the number ofthe components uε,k such that for uε,k(x) := uε,k(ε

1/3x), we have

˜E∞(uε,k) ≥ ( f ∗ + δ)

∫

R3uε,k dx . (7.3)

By (3.8), (3.12) and the arguments in the proof of Proposition 5.1 we have, as ε → 0,

λ f ∗ = ε−4/3Eε(uε) + oε(1) ≥ ε1/3Nε∑

k=1

˜E∞(uε,k) + oε(1)

≥ ε1/3(

( f ∗ + δ)

Nε,δ∑

k=1

∫

R3uε,k dx + f ∗

Nε∑

k=Nε,δ+1

∫

R3uε,k dx

)

+ oε(1)

≥ λ f ∗ + c δNε,δ ε1/3 + oε(1), (7.4)

where we suitably ordered all uε,k and included a possibility that the range of summationis empty in either of the two sums. Hence, Nε,δ = o(ε−1/3), and by (ii) it follows thatNε,δ = o(Nε) for all δ > 0. This implies that for every δ > 0 there is εδ > 0 and a collec-tion of indices Iεδ satisfying (#Iεδ )/Nεδ → 1 such that ˜E∞(uεδ,k)/

∫

R3 uεδ,k dx → f ∗uniformly in k ∈ Iεδ as δ → 0. By (ii), for every sequence of δn → 0 and every choice ofkn ∈ Iεδn

the sequence un := uεδn ,kn is supported in BR(0) for some R > 0 universal andequibounded in BV (R3). Hence, upon extraction of a subsequence we have un → u inL1(R3) with m := ∫

R3 u dx > 0. At the same time, by lower semicontinuity of ˜E∞ wealso have ˜E∞(u)/m ≤ f ∗. Then, by Theorem 4.15 the latter is, in fact, an equality, andso un(x) := u(λnx) with λn := (m−1

∫

R3 uε,k dx)1/3 → 1, is a minimizing sequencefor ˜E∞ over Ã∞(m) (cf. (4.17)). Thus, u is a minimizer of ˜E∞ over Ã∞(m). Again, byTheorem 4.15 we then have m ∈ I∗.


Acknowledgements. The work of CBM was partly supported by NSF via Grants DMS-0908279 and DMS-1313687. The work of MN was partly supported by the Italian CNR-GNAMPA and by the University of Pisavia Grant PRA-2015-0017. CBM gratefully acknowledges the hospitality of the University of Heidelberg. HKand CBM gratefully acknowledge the hospitality of the Max Planck Institute for Mathematics in the Sciences,and both CBM and MN gratefully acknowledge the hospitality of Mittag-Leffler Institute, where part of thiswork was completed.

A. Appendix

We recall that by the Riesz–Fischer theorem, the space of signed RadonmeasuresM(T)

is embedded in the space of distributions via the identification

〈ϕ,μ〉 :=∫

T

ϕdμ ∀ϕ ∈ C∞(T). (A.1)

On the other hand, any measure μ ∈ M+(T) ∩ H′ (recall the definition in (2.12)) canbe extended by continuity to an element of the dual space H′, which we still denote byμ, such that

∫

T

ϕ dμ = H〈ϕ,μ〉H′ ∀ϕ ∈ H ∩ C0(T). (A.2)

Lemma A.1. Let μ ∈ M+(T) ∩ H′ and u ∈ H. Then, up to taking the preciserepresentative, u belongs to L1(T, dμ) and

H〈u, μ〉H′ =∫

T

u dμ. (A.3)

Proof. The result follows as in [8, Theorem 1]. For the reader’s convenience we includea simple alternative proof here. Since u ∈ H, by [21, Section 4.8: Theorem 1] we canidentify u with its precise representative and find a sequence uk ∈ H∩C0(T) such thatuk → u inH, and

uk(x) → u(x) for all x �∈ N , (A.4)

where N ⊂ T is a set of zero inner capacity, that is, for any compact set K ⊂ N thereexists a sequence ϕn ∈ H ∩ C0(T) such that ϕn → 0 in H and ϕn = 1 on K . Sinceμ ∈ H′ we have μ(K ) = 0 for all compact K ⊂ N , so that

μ(N ) = supK⊂N

μ(K ) = 0. (A.5)

Since the functions uk are continuous for all k ∈ N, we have

H〈uk, μ〉H′ =∫

T

ukdμ, (A.6)

Therefore, by (A.2) we get

H〈|uk − uk′ | − αk,k′ , μ〉H′ =∫

T

|uk − uk′ |dμ − αk,k′ μ(T), (A.7)

for all k′ ∈ N, where

αk,k′ :=∫

T

|uk − uk′ | dx . (A.8)


It then follows

‖uk − uk′ ‖L1(T,dμ) ≤ ‖|uk − uk′ | − αk,k′ ‖H‖μ‖H′ + μ(T)‖uk − uk′ ‖L1(T)

= ‖∇(uk − uk′)‖L2(T)‖μ‖H′ + μ(T)‖uk − uk′ ‖L1(T). (A.9)

Since uk is a Cauchy sequence inH, hence also in L1(T), from (A.9) it follows thatuk is a Cauchy sequence in L1(T, dμ) and, therefore, converges to some u ∈ L1(T, dμ).In fact, passing to a subsequence and using (A.4) and (A.5), we have u(x) = u(x) forμ-a.e. x ∈ T. Therefore, from (A.6) we get

H〈u, μ〉H′ = limk→∞ H〈uk, μ〉H′ = lim

k→∞

∫

T

uk dμ =∫

T

u dμ, (A.10)

which concludes the proof. ��The following lemma characterizes themeasures in terms of the Coulombic potential,

see [32, Lemma 3.2] for a related result.

Lemma A.2. Let μ ∈ M+(T) ∩ H′, and let G : T → (−∞,+∞] be the uniquedistributional solution of (2.6) with G(0) = +∞. Then the function

v(x) :=∫

T

G(x − y) dμ(y) x ∈ T (A.11)

belongs toH and solves

−∫

T

v�ϕ dx =∫

T

ϕ dμ ∀ϕ ∈ C∞(T) ∩ H. (A.12)

Moreover v ∈ L1(T, dμ) and

∫

T

∫

T

G(x − y) dμ(x) dμ(y) =∫

T

v dμ =∫

T

|∇v|2 dx . (A.13)

Proof. By the definition of G and the fact that G ∈ L1(T), the function v belongs toL1(T), solves (A.12) and has zero average on T. On the other hand, by (2.12) one candefine a functional Tμ ∈ H′ such that Tμ(ϕ) = ∫

Tϕ dμ for every ϕ ∈ C∞(T) ∩ H.

Therefore, by Riesz Representation Theorem there exists v ∈ H such that

−∫

T

v�ϕ dx = 〈ϕ, v〉H = −∫

T

v�ϕ dx ∀ϕ ∈ C∞(T) ∩ H. (A.14)

Thus, since � is a one-to-one map from C∞(T) ∩ H to itself, we conclude that v = v

almost everywhere with respect to the Lebesgue measure on T and, hence, v ∈ H.Let now ρ ∈ C∞(T) be a radial symmetric-decreasing mollifier supported on

B1/8(0), let ρn(x) := n3ρ(nx), so that ρn → δ0 in D′(T), and let fn ∈ C∞(T) bedefined as

fn(x) :=∫

T

ρn(x − y) dμ(y) x ∈ T. (A.15)


Then, if the measuresμn ∈ M+(T)∩H′ are such that dμn = fn dx , we have Tμn → Tμ

in H′ and μn ⇀ μ in M(T). Letting also vn(x) := ∫

TG(x − y) dμn(y), we observe

that vn → v ∈ H, and μn ⊗ μn ⇀ μ ⊗ μ inM(T × T). For all M > 0, we then get∫

T

∫

T

GM (x − y) dμ(x) dμ(y) = limn→∞

∫

T

∫

T

GM (x − y) dμn(x) dμn(y),(A.16)

where we set GM (x) := min(G(x), M) ∈ C(T). By Monotone Convergence Theoremwe also have

∫

T

∫

T

G(x − y) dμ(x) dμ(y) = limM→∞

∫

T

∫

T

GM (x − y) dμ(x) dμ(y). (A.17)

Recalling (A.16), it then follows∫

T

∫

T

G(x − y) dμ(x) dμ(y) = limM→∞ lim

n→∞

∫

T

∫

T

GM (x − y) dμn(x) dμn(y)

≤ limn→∞

∫

T

∫

T

G(x − y) dμn(x) dμn(y)

= limn→∞

∫

T

vn dμn = limn→∞ ‖vn‖2H = ‖v‖2H. (A.18)

Together with the fact that G is bounded from below, by Fubini–Tonelli theorem thisimplies that v ∈ L1(T, dμ), with ‖v‖L1(T,dμ) ≤ ‖v‖2H.

It remains to prove (A.13). We reason as in [42, Theorem 1.11] and pass to the limit,as n → ∞, in the equality

∫

T

vn dμn =∫

T

|∇vn|2 dx, (A.19)

which holds for all n ∈ N. Notice that the right-hand side of (A.19) converges sincevn → v inH, so that

limn→∞

∫

T

|∇vn|2 dx =∫

T

|∇v|2 dx . (A.20)

In order to pass to the limit in the left-hand side of (A.19), we write∫

T

vn dμn =∫

T

∫

T

G(x − y) dμn(x)dμn(y) =∫

T

∫

T

Gn(x − y) dμ(x)dμ(y),

(A.21)

where we set

Gn(x) :=∫

T

G(x − y)ρn(y) dy, (A.22)

ρn(x) :=∫

T

ρn(x − y)ρn(y) dy. (A.23)

We claim that there exists C > 0 such that

|Gn(x)| ≤ C (1 + |G(x)|) (A.24)


for all x ∈ T. Indeed, we can write G = �# + R as in (5.15). Letting

�#n(x) :=

∫

T

�#(x − y)ρn(y) dy and Rn(x) :=∫

T

R(x − y)ρn(y) dy, (A.25)

we have that Rn → R uniformly as n → ∞. Moreover, since�#,�#n and ρn are periodic

when viewed as functions on R3, rewriting the integrals as integrals over subsets of R

3

and applying Newton’s Theorem we get

�#n(x)

�#(x)= 4π |x |

∫

B1/4(0)�#(x − y)ρn(y) dy = |x |

∫

B1/4(0)

ρn(y)

|x − y| dy

=∫

B|x |(0)ρn(y) dy + |x |

∫

B1/4(0)\B|x |(0)

ρn(y)

|y| dy

≤∫

B1/4(0)ρn(y) dy = 1 for all |x | <

1

4. (A.26)

Since also

�#n(x)

�#(x)= 4π |x |

∫

B1/8(0)�#(x − y)ρn(y) dy ≤ C for all

1

4≤ |x | ≤

√3

2,

(A.27)

this proves (A.24).From the fact that Gn(x) → G(x) for all x ∈ T, by (A.24) and the Dominated

Convergence Theorem we get

limn→∞

∫

T

vn dμn = limn→∞

∫

T

∫

T

Gn(x − y) dμ(x)dμ(y)

=∫

T

∫

T

G(x − y) dμ(x)dμ(y) =∫

T

v dμ. (A.28)

From (A.19), (A.20) and (A.28) we obtain (A.13). ��Lemma A.3. Let G be as in Lemma A.2 and let μ ∈ M+(T) satisfy (2.13). Thenμ ∈ M+(T) ∩ H′.Proof. Let ϕ ∈ C0(T) ∩ H. Using the same notation and arguments as in the proof ofLemma A.2, with the help of Cauchy–Schwarz inequality we obtain

∫

T

ϕ dμ = limn→∞

∫

T

ϕ dμn = limn→∞

∫

T

∇ϕ · ∇vn dx

≤ ‖ϕ‖H limn→∞

(∫

T

∫

T

G(x − y) dμn(x) dμn(y)

) 12

= ‖ϕ‖H limn→∞

(∫

T

∫

T

Gn(x − y) dμ(x) dμ(y)

) 12

= ‖ϕ‖H(∫

T

∫

T

G(x − y) dμ(x) dμ(y)

) 12

, (A.29)

which yields the inequality in (2.12). ��


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Communicated by H.-T. Yau

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Low Density Phases in a Uniformly Charged Liquidmuratov/ndrop3dok.pdf · Low Density Phases in a Uniformly Charged Liquid ... which we will also refer to as the “self-energy problem”,